Stochastic Integration and Differential Equations

Stochastic Mechanics Random Media Signal Processing and Image Synthesis Mathematical Economics and Finance Stochastic Optimization Stochastic Control Stochastic Models in Life Sciences Applications of Mathematics Stochastic Modelling and Applied Probability 21 Edited by B. Rozovskii M. Yor Advisory Board D. Dawson D.Geman G. Grimmett I. Karatzas F. Kelly Y. Le Jan B.0ksendal E. Pardoux G. Papanicolaou Springer Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Applications of Mathematics 1 Fleming/Rishel, Deterministic and Stochastic Optimal Control (1975) 2 Marchuk, Methods ofNumerical Mathematics 1975, 2nd. ed. 1982) 3 Balakrishnan, Applied Functional Analysis (1976, 2nd. ed. 1981) 4 Borovkov, Stochastic Processes in Queueing Theory (1976) 5 Liptser/Shiryaev, Statistics ofRandom Processes I: General Theory (19n md. ed. 2001) 6 Liptser/Shiryaev, Statistics of Random Processes II: Applications (1978, 2nd. ed. 2001) 7 Vorob'ev, Game Theory: Lectures for Economists and Systems Scientists (1977) 8 Shiryaev, Optimal Stopping Rules (1978) 9 Ibragimov/Rozanov, Gaussian Random Processes (1978) 10 Wonham, Linear Multivariable Control: A Geometric Approach (1979, 2nd. ed. 1985) 11 Hida, Brownian Motion (1980) 12 Hestenes, Conjugate Direction Methods in Optimization (1980) 13 Kallianpur, Stochastic Filtering Theory (1980) 14 Krylov, Controlled Diffusion Processes (1980) 15 Prabhu, Stochastic Storage Processes: Queues, Insurance Risk, and Dams (1980) 16 IbragimovlHas'minskii, Statistical Estimation: Asymptotic Theory (1981) 17 Cesari, Optimization: Theory and Applications (1982) 18 Elliott, Stochastic Calculus and Applications (1982) 19 MarchuklShaidourov, Difference Methods and Their Extrapolations (1983) 20 Hijab, Stabilization of Control Systems (1986) 21 Protter, Stochastic Integration and Differential Equations (1990, md. ed. 2003) 22 Benveniste/Metivier/Priouret, Adaptive Algorithms and Stochastic Approximations (1990) 23 Kloeden/Platen, Numerical Solution of Stochastic Differential Equations (1992, corr. 3rd printing 1999) 24 KushnerlDupuis, Numerical Methods for Stochastic Control Problems in Continuous Time (1992) 25 Fleming/Soner, Controlled Markov Processes and Viscosity Solutions (1993) 26 BaccellilBremaud, Elements ofQueueing Theory (1994, 2nd ed. 2003) 27 Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods (1995, 2nd. ed. 2003) 28 Kalpazidou, Cycle Representations ofMarkov Processes (1995) 29 ElliottlAggoun/Moore, Hidden Markov Models: Estimation and Control (1995) 30 Hermlndez-Lerma/Lasserre, Discrete-Time Markov Control Processes (1995) 31 Devroye/GyorfilLugosi, A Probabilistic Theory of Pattern Recognition (1996) 32 Maitra/Sudderth, Discrete Gambling and Stochastic Games (1996) 33 Embrechts/Kliippelberg/Mikosch, Modelling Extremal Events for Insurance and Finance (1997, corr. 4th printing 2003) 34 Dullo, Random Iterative Models (1997) 35 Kushner/Yin, Stochastic Approximation Algorithms and Applications (1997) 36 MusielalRutkowski, Martingale Methods in Financial Modelling (1997) 37 Yin, Continuous-Time Markov Chains and Applications (1998) 38 Dembo/Zeitouni, Large Deviations Techniques and Applications (1998) 39 Karatzas, Methods ofMathematical Finance (1998) 40 Fayolle/lasnogorodskilMalyshev, Random Walks in the Quarter-Plane (1999) 41 Aven/Jensen, Stochastic Models in Reliability (1999) 42 Hernandez-Lerma/Lasserre, Further Topics on Discrete-Time Markov Control Processes (1999) 43 Yong/Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations (1999) 44 Serfozo, Introduction to Stochastic Networks (1999) 45 Steele, Stochastic Calculus and Financial Applications (2001) 46 Chen/Yao, Fundamentals of Queuing Networks: Performance, Asymptotics, and Optimization (2001) 47 Kushner, Heavy Traffic Analysis of Controlled Queueing and Communications Networks (2001) 48 Fernholz, Stochastic Portfolio Theory (2002) 49 Kabanov/Pergamenshchikov, Two-Scale Stochastic Systems (2003) 50 Han, Information-Spectrum Methods in Information Theory (2003) (continued after index) Philip E. Protter Stochastic Integration and Differential Equations Second Edition , Springer ......--------- Author Philip E. Protter Cornell University School of Operations Res. and Industrial Engineering Rhodes Hall 14853 Ithaca, NY USA e-mail: [email protected] Managing Editors B. Rozovskii Center for Applied Mathematical Sciences University ofSouthern California 1042 West 36th Place, Denney Research Building 308 Los Angeles, CA 90089, USA M. Yor Universite de Paris VI Laboratoire de Probabilites et Modeles Aleatoires 175, rue du Chevaleret 75013 Paris, France Mathematics Subject Classification (2000): PRIMARY: 6OH05, 60HI0, 60H20 SECONDARY: 60G07, 60G17, 60G44, 60G51 Cover pattern by courtesy of Rick Durrett (Cornell University, Ithaca) Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographie information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISSN 0172-4568 ISBN 3-540-00313-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de C Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg Typesetting by the author using a Springer TEX macro package Printed on acid-free paper 41/3142D8-54321O To Diane and Rachel Preface to the Second Edition It has been thirteen years since the first edition was published, with its subtitle "a new approach." While the book has had some success, there are still almost no other books that use the same approach. (See however the recent book by K. Bichteler [15].) There are nevertheless of course other extant books, many of them quite good, although the majority still are devoted primarily to the case of continuous sample paths, and others treat stochastic integration as one of many topics. Examples of alternative texts which have appeared since the first edition of this book are: [32], [44], [87], [110], [186], [180], [208], [216], and [226]. While the subject has not changed much, there have been new developments, and subjects we thought unimportant in 1990 and did not include, we now think important enough either to include or to expand in this book. The most obvious changes in this edition are that we have added exercises at the end of each chapter, and we have also added Chap. VI which intro- duces the expansion of filtrations. However we have also completely rewritten Chap. III. In the first edition we followed an elementary approach which was P. A. Meyer's original approach before the methods of Doleans-Dade. In or- der to remain friends with Freddy Delbaen, and also because we now agree with him, we have instead used the modern approach of predictability rather than naturality. However we benefited from the new proof of the Doob-Meyer Theorem due to R. Bass, which ultimately uses only Doob's quadratic martin- gale inequality, and in passing reveals the role played by totally inaccessible stopping times. The treatment of Girsanov's theorem now includes the case where the two probability measures are not necessarily equivalent, and we include the Kazamaki-Novikov theorems. We have also added a section on compensators, with examples. In Chap. IV we have expanded our treatment of martingale representation to include the Jacod-Yor Theorem, and this has allowed us to use the Emery-Azema martingales as a class of examples of mar- tingales with the martingale representation property. Also, largely because of the Delbaen-Schachermayer theory of the fundamental theorems of mathe- matical finance, we have included the topic of sigma martingales. In Chap. V VIII Preface to the Second Edition we added a section which includes some useful results about the solutions of stochastic differential equations, inspired by the review of the first edition by E. Pardoux [191]. We have also made small changes throughout the book; for instance we have included specific examples of Levy processes and their corresponding Levy measures, in Sect. 4 of Chap. 1. The exercises are gathered at the end of the chapters, in no particular order. Some of the (presumed) harder problems we have designated with a star (*), and occasionally we have used two stars (**). While of course many of the problems are of our own creation, a significant number are theorems or lemmas taken from research papers, or taken from other books. We do not attempt to ascribe credit, other than listing the sources in the bibliography, primarily because they have been gathered over the past decade and often we don't remember from where they came. We have tried systematically to refrain from relegating a needed lemma as an exercise; thus in that sense the exercises are independent from the text, and (we hope) serve primarily to illustrate the concepts and possible applications of the theorems. Last, we have the pleasant task of thanking the numerous people who helped with this book, either by suggesting improvements, finding typos and mistakes, alerting me to references, or by reading chapters and making com- ments. We wish to thank patient students both at Purdue University and Cornell University who have been subjected to preliminary versions over the years, and the following individuals: C. Benes, R. Cont, F. Diener, M. Di- ener, R. Durrett, T. Fujiwara, K. Giesecke, L. Goldberg, R. Haboush, J. Ja- cod, H. Kraft, K. Lee, J. Ma, J. Mitro, J. Rodriguez, K. Schiirger, D. Sezer, J. A. Trujillo Ferreras, R. Williams, M. Yor, and Yong Zeng. Th. Jeulin, K. Shimbo, and Yan Zeng gave extraordinary help, and my editor C. Byrne gives advice and has patience that is impressive. Over the last decade I have learned much from many discussions with Darrell Duffie, Jean Jacod, Tom Kurtz, and Denis Talay, and this no doubt is reflected in this new edition. Finally, I wish to give a special thanks to M. Kozdron who hastened the ap- pearance of this book through his superb help with M\'IEX, as well as his own advice on all aspects of the book. Ithaca, NY August 2003 Philip Protter Preface to the First Edition The idea of this book began with an invitation to give a course at the Third Chilean Winter School in Probability and Statistics, at Santiago de Chile, in July, 1984. Faced with the problem of teaching stochastic integration in only a few weeks, I realized that the work of C. Dellacherie [42] provided an outline for just such a pedagogic approach. I developed this into a series of lectures (Protter [201]), using the work of K. Bichteler [14], E. Lenglart [145] and P. Protter [202], as well as that of Dellacherie. I then taught from these lecture notes, expanding and improving them, in courses at Purdue University, the University of Wisconsin at Madison, and the University of Rouen in France. I take this opportunity to thank these institutions and Professor Rolando Rebolledo for my initial invitation to Chile. This book assumes the reader has some knowledge of the theory of stochas- tic processes, including elementary martingale theory. While we have recalled the few necessary martingale theorems in Chap. I, we have not provided proofs, as there are already many excellent treatments of martingale the- ory readily available (e.g., Breiman [23], Dellacherie-Meyer [45, 46], or Ethier- Kurtz [71]). There are several other texts on stochastic integration, all of which adopt to some extent the usual approach and thus require the general theory. The books of Elliott [63], Kopp [130]' Metivier [158], Rogers-Williams [210] and to a much lesser extent Letta [148] are examples. The books of McK- ean [153], Chung-Williams [32], and Karatzas-Shreve [121] avoid the general theory by limiting their scope to Brownian motion (McKean) and to contin- uous semimartingales. Our hope is that this book will allow a rapid introduction to some of the deepest theorems of the subject, without first having to be burdened with the beautiful but highly technical "general theory of processes." Many people have aided in the writing of this book, either through dis- cussions or by reading one of the versions of the manuscript. I would like to thank J. Azema, M. Barlow, A. Bose, M. Brown, C. Constantini, C. Dellache- rie, D. Duffie, M. Emery, N. Falkner, E. Goggin, D. Gottlieb, A. Gut, S. He, J. Jacod, T. Kurtz, J. de Sam Lazaro, R. Leandre, E. Lenglart, G. Letta, X Preface to the First Edition S. Levantal, P. A. Meyer, E. Pardoux, H. Rubin, T. Sellke, R. Stockbridge, C. Stricker, P. Sundar, and M. Yor. I would especially like to thank J. San Mar- tin for his careful reading of the manuscript in several of its versions. Svante Janson read the entire manuscript in several versions, giving me support, encouragement, and wonderful suggestions, all of which improved the book. He also found, and helped to correct, several errors. I am extremely grateful to him, especially for his enthusiasm and generosity. The National Science Foundation provided partial support throughout the writing of this book. I wish to thank Judy Snider for her cheerful and excellent typing of several versions of this book. Philip Protter Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 I Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Basic Definitions and Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 The Poisson Process and Brownian Motion 12 4 Levy Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19 5 Why the Usual Hypotheses? 34 6 Local Martingales 37 7 Stieltjes Integration and Change of Variables. . . . . . . . . . . . . . .. 39 8 NaIve Stochastic Integration Is Impossible. . . . . . . . . . . . . . . . .. 43 Bibliographic Notes . . . . .. 44 Exercises for Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45 II Semimartingales and Stochastic Integrals 51 1 Introduction to Semimartingales. . . . . . . . . . . . . . . . . . . . . . . . . .. 51 2 Stability Properties of Semimartingales . . . . . . . . . . . . . . . . . . . .. 52 3 Elementary Examples of Semimartingales. . . . . . . . . . . . . . . . . .. 54 4 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56 5 Properties of Stochastic Integrals , 60 6 The Quadratic Variation of a Semimartingale , 66 7 Ito's Formula (Change of Variables) " 78 8 Applications of Ito's Formula 84 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92 Exercises for Chapter II 94 III Semimartingales and Decomposable Processes 101 1 Introduction 101 2 The Classification of Stopping Times 103 3 The Doob-Meyer Decompositions 105 4 Quasimartingales 116 XII Contents 5 Compensators 118 6 The Fundamental Theorem of Local Martingales 124 7 Classical Semimartingales 127 8 Girsanov's Theorem 131 9 The Bichteler-Dellacherie Theorem 143 Bibliographic Notes 147 Exercises for Chapter III 147 IV General Stochastic Integration and Local Times , 153 1 Introduction 153 2 Stochastic Integration for Predictable Integrands 153 3 Martingale Representation 178 4 Martingale Duality and the Jacod-Yor Theorem on Martingale Representation 193 5 Examples of Martingale Representation 200 6 Stochastic Integration Depending on a Parameter 205 7 Local Times 210 8 Azema's Martingale 227 9 Sigma Martingales 233 Bibliographic Notes 235 Exercises for Chapter IV 236 V Stochastic Differential Equations 243 1 Introduction 243 2 The HP Norms for Semimartingales 244 3 Existence and Uniqueness of Solutions 249 4 Stability of Stochastic Differential Equations 257 5 Fisk-Stratonovich Integrals and Differential Equations 270 6 The Markov Nature of Solutions 291 7 Flows of Stochastic Differential Equations: Continuity and Differentiability 301 8 Flows as Diffeomorphisms: The Continuous Case 310 9 General Stochastic Exponentials and Linear Equations 321 10 Flows as Diffeomorphisms: The General Case 328 11 Eclectic Useful Results on Stochastic Differential Equations 338 Bibliographic Notes 347 Exercises for Chapter V 349 VI Expansion of Filtrations 355 1 Introduction 355 2 Initial Expansions 356 3 Progressive Expansions 369 4 Time Reversal 377 Bibliographic Notes 383 Exercises for Chapter VI 384 Contents XIII References 389 Symbol Index 403 Subject Index 407 Introduction In this book we present a new approach to the theory of modern stochastic integration. The novelty is that we define a semimartingale as a stochastic pro- cess which is a "good integrator" on an elementary class of processes, rather than as a process that can be written as the sum of a local martingale and an adapted process with paths of finite variation on compacts: This approach has the advantage over the customary approach of not requiring a close analysis of the structure of martingales as a prerequisite. This is a significant advantage because such an analysis of martingales itself requires a highly technical body of knowledge known as "the general theory of processes." Our approach has a further advantage of giving traditionally difficult and non-intuitive theorems (such as Stricker's Theorem) transparently simple proofs. We have tried to capitalize on the natural advantage of our approach by systematically choos- ing the simplest, least technical proofs and presentations. As an example we have used K. M. Roo's proofs of the Doob-Meyer decomposition theorems in Chap. III, rather than the more abstract but less intuitive Doleans-Dade measure approach. In Chap. I we present preliminaries, including the Poisson process, Brown- ian motion, and Levy processes. Naturally our treatment presents those prop- erties of these processes that are germane to stochastic integration. In Chap. II we define a semimartingale as a good integrator and establish many of its properties and give examples. By restricting the class of integrands to adapted processes having left continuous paths with right limits, we are able to give an intuitive Riemann-type definition of the stochastic integral as the limit of sums. This is sufficient to prove many theorems (and treat many applications) including a change of variables formula ("Ito's formula"). Chapter III is devoted to developing a minimal amount of "general the- ory" in order to prove the Bichteler-Dellacherie Theorem, which shows that our "good integrator" definition of a semimartingale is equivalent to the usual one as a process X having a decomposition X = M + A, into the sum of a local martingale M and an adapted process A having paths of finite variation on compacts. Nevertheless most of the theorems covered en route (Doob- 2 Introduction Meyer, Meyer-Girsanov) are themselves key results in the theory. The core of the whole treatment is the Doob-Meyer decomposition theorem. We have followed the relatively recent proof due to R. Bass, which is especially simple for the case where the martingale jumps only at totally inaccessible stopping times, and in all cases uses no mathematical tool deeper than Doob's quadratic martingale inequality. This allows us to avoid the detailed treatment of nat- ural processes which was ubiquitous in the first edition, although we still use natural processes from time to time, as they do simplify some proofs. Using the results of Chap. III we extend the stochastic integral by continu- ity to predictable integrands in Chap. IV, thus making the stochastic integral a Lebesgue-type integral. We use predictable integrands to develop a theory of martingale representation. The theory we develop is an £2 theory, but we also prove that the dual of the martingale space Hi is BMO and then prove the Jacod-Yor Theorem on martingale representation, which in turn allows us to present a class of examples having both jumps and martingale representation. We also use predictable integrands to give a presentation of semimartingale local times. Chapter V serves as an introduction to the enormous subject of stochastic differential equations. We present theorems on the existence and uniqueness of solutions as well as stability results. Fisk-Stratonovich equations are pre- sented, as well as the Markov nature of the solutions when the differentials have Markov-type properties. The last part of the chapter is an introduction to the theory of flows, followed by moment estimates on the solutions, and other minor but useful results. Throughout Chap. V we have tried to achieve a balance between maximum generality and the simplicity of the proofs. Chapter VI provides an introduction to the theory of the expansion of fil- trations (known as "grossissements de filtrations" in the French literature). We present first a theory of initial expansions, which includes Jacod's Theorem. Jacod's Theorem gives a sufficient condition for semimartingales to remain semimartingales in the expanded filtration. We next present the more diffi- cult theory of progressive expansion, which involves expanding filtrations to turn a random time into a stopping time, and then analyzing what happens to the semimartingales of the first filtration when considered in the expanded filtration. Last, we give an application of these ideas to time reversal. I Preliminaries 1 Basic Definitions and Notation We assume as given a complete probability space (0, F, P). In addition we are given a filtration (Ft)o:s;t:s;oo' By a filtration we mean a family of (I-algebras (Ft)ou>t{T < u}, any c: > 0, we have {T ~ t} E nu>t Fu = Ft , so T is a stopping time. For the converse, {T < t} = Ut>e>o{T ~ t - c:}, and {T ~ t - c:} E Ft-€, hence also in Ft. D A stochastic process X on (0, F, P) is a collection of JR-valued or JRd_ valued random variables (Xt)o:s;t 4 I Preliminaries IfX and Yare modifications there exists a null set, Nt, such that if w ~ Nt, then Xt(w) = yt(w). The null set Nt depends on t. Since the interval [0,(0) is uncountable the set N = Uooo ytJw) = yt(w). Since P(M) = 0, X and Yare indistinguishable. D Corollary. Let X and Y be two stochastic processes which are cadlag. If X is a modification of Y, then X and Yare indistinguishable. CadUtg processes provide natural examples of stopping times. Definition. Let X be a stochastic process and let A be a Borel set in R Define T(w) = inf{t > °:X t E A}. Then T is called a hitting time of A for X. Theorem 3. Let X be an adapted cadlag stochastic process, and let A be an open set. Then the hitting time of A is a stopping time. Proof By Theorem 1 it suffices to show that {T < t} E :Ft , °::::; t < 00. But {T < t} = U {Xs E A}, sEQn[O,t) since A is open and X has right continuous paths. Since {Xs E A} = X;l(A) E :Fs , the result follows. D 1 Basic Definitions and Notation 5 Theorem 4. Let X be an adapted cadlag stochastic process, and let A be a closed set. Then the random variable T(w) = inf{t > 0: Xt(w) E A or Xt-(w) E A} is a stopping time. Proof By Xt-(w) we mean lims->t,s l. The O"-algebra F t can be thought of as representing all (theoretically) ob- servable events up to and including time t. We would like to have an analogous notion of events that are observable before a random time. Definition. Let T be a stopping time. The stopping time O"-algebra FT is defined to be The previous definition is not especially intuitive. However it does well represent "knowledge" up to time T, as the next theorem illustrates. Theorem 6. Let T be a finite stopping time. Then F T is the smallest 0"- algebra containing all cadlag processes sampled at T. That is, FT = O"{XT; X all adapted cadlag processes}. Proof Let 9 = O"{XT;X all adapted dldlag processes}. Let A E FT. Then X t = lAl{t~T} 1 is a cadlag process, and X T = lA. Hence A E g, and FT C g. 1 lA is the indicator function of A: lA(w) = {I, 0, WEA, w~ A. 6 I Preliminaries Next let X be an adapted cfl,dlag process. We need to show X T is FT measurable. Consider X (s, w) as a function from [0, (0) x 0 into R Construct cp : {T ::; t} ~ [0, (0) x 0 by cp(w) = (T(w),w). Then since X is adapted and cadlag, we have XT = X 0cp is a measurable mapping from ({T ::; t}, Ftn{T ::; t}) into (IR, 8), where 8 are the Borel sets of R Therefore {w : X(T(w),w) E B} n {T::; t} is in F t , and this implies X T E FT. Therefore 9 eFT. D We leave it to the reader to check that if S ::; T a.s., then Fs eFT, and the less obvious (and less important) fact that Fs nFT = FSIIT. If X and Yare cadlag, then X t = yt a.s. each t implies that X and Yare indistinguishable, as we have already noted. Since fixed times are stopping times, obviously if XT = YT a.s. for each finite stopping time T, then X and Yare indistinguishable. If X is cadlag, let ~X denote the process ~Xt = X t - X t-. Then ~X is not cadlag, though it is adapted and for a.a. w, t ~ ~Xt = a except for at most countably many t. We record here a useful result. Theorem 7. Let X be adapted and cddldg. If ~XTI{T a} is countable a.s. since X is cadlag. Moreover 00 I {t : I~Xtl > a} = U{t : I~Xtl > -} n n=l and the set {t : I~Xtl > lin} must be finite for each n, since to < 00. Using Theorem 4 we define stopping times for each n inductively as follows: Tn,l = inf{t > a: I~Xtl > ..!:.} n ITn,k = inf{t > Tn,k-l : I~Xtl > -}. n Then Tn,k > Tn,k-l a.s. on {Tn,k-l < oo}. Moreover, {1~Xtl > a} = U{I~XTn,d{Tn,k a}, n,k where the right side of the equality is a countable union. The result follows. D Corollary. Let X and Y be adapted and cadlag. If for each stopping time T, ~XTI{T 2 Martingales 7 A much more general version of Theorem 7 is true, but it is a very deep result which uses Meyer's "section theorems," and we will not have need of it. See, for example, Dellacherie [41J or Dellacherie-Meyer [45J. A fundamental theorem of measure theory that we will need from time to time is known as the MonOtone Class Theorem. Actually there are several such theorems, but the one given here is sufficient for our needs. Definition. A monotone vector space H on a space 0 is defined to be the collection of bounded, real-valued functions f on 0 satisfying the three conditions: (i) H is a vector space over lR; (ii) In E H (i.e., constant functions are in H); and (iii) if (fn)n"21 C H, and 0 ::; It ::; h ::; ... ::; fn ::; ... , and limn--->oo fn = f, and f is bounded, then f E H. Definition. A collection M of real functions defined on a space 0 is said to be multiplicative if f, gEM implies that f gEM. For a collection of real-valued functions M defined on 0, we let a{M} denote the space of functions defined on 0 which are measurable with respect to the a-algebra on 0 generated by U-1(A); A E B(lR), f EM}. Theorem 8 (Monotone Class Theorem). Let M be a multiplicative class of bounded real-valued functions defined on a space 0, and let A = a{M}. If H is a monotone vector space containing M, then H contains all bounded, A measurable functions. Theorem 8 is proved in Dellacherie-Meyer [45, page 14J with the additional hypothesis that H is closed under uniform convergence. This extra hypothesis is unnecessary, however, since every monotone vector space is closed under uniform convergence. (See Sharpe [215, page 365J.) 2 Martingales In this section we give, mostly without proofs, only the essential results from the theory of continuous time martingales. The reader can consult any of a large number of texts to find excellent proofs; for example Dellacherie- Meyer [46], or Ethier-Kurtz [71]. Also, recall that we will always assume as given a filtered, complete probability space (0, F, IF, P), where the filtration IF = (Ft}o$;t$;oo is assumed to be right continuous. Definition. A real-valued, adapted process X = (Xt)O$;t 8 I Preliminaries Note that martingales are only defined on [0, (0); that is, for finite t and not t = 00. It is often possible to extend the definition to t = 00. Definition. A martingale X is said to be closed by a random variable Y if E{!YI} < 00 and X t = E{YIFt}, 0 :::; t < 00. A random variable Y closing a martingale is not necessarily unique. We give a sufficient condition for a martingale to be closed (as well as a construc- tion for closing it) in Theorem 12. Theorem 9. Let X be a supermartingale. The function t f--> E{Xt} is right continuous if and only if there exists a modification Y of X which is cadlag. Such a modification is unique. By uniqueness we mean up to indistinguishability. Our standing assump- tion that the "usual hypotheses" are satisfied is used implicitly in the state- ment of Theorem 9. Also, note that the process Y is, of course, also a super- martingale. Theorem 9 is proved using Doob's upcrossing inequalities. If X is a martingale then t f--> E{Xtl is constant, and hence it has a right continuous modification. Corollary. If X = (Xt)o: 2 Martingales 9 (iii) There exists a positive, increasing, convex function G(x) defined on [0,(0) such that limx-+ oo G~x) = +00 and sUPa E{Go Wa I} < 00. The assumption that G is convex is not needed for the implications (iii) '* (ii) and (iii) '* (i). Theorem 12. Let X be a right continuous martingale which is uniformly integrable. Then Y = limt-+oo X t a.s. exists, E{IYI} < 00, and Y closes X as a martingale. Theorem 13. Let X be a (right continuous) martingale. Then (Xt)t>o is uniformly integrable if and only if Y = limt-+oo Xt exists a.s., E{IYI} 10 I Preliminaries Theorem 17. Let X be a right continuous supermartingale (resp. martin- gale), and let 5 and T be two bounded stopping times such that 5 ::; T a.s. Then Xs and X T are integrable and Xs 2: E{XTIFs} a.s. (resp. =). If T is a stopping time, then so is t 1\ T = min(t, T), for each t ::::: O. Definition. Let X be a stochastic process and let T be a random time. X T is said to be the process stopped at T if Xl' = XtI\T. Note that if X is adapted and dtdlag and if T is a stopping time, then xi = Xtl\T = X t1{t 2 Martingales 11 Proof Let yt = E{YIFt }. Then yT is a uniformly integrable martingale and YSI\T = YI = E{YTIFs} = E{E{YIFT }IFs}. Interchanging the roles of T and S yields YSI\T = Y,J = E{YsIFT} = E{E{YIFs}IFT}. Finally, E{YIFsI\T} = YSI\T. o The next inequality is elementary, but indispensable. Theorem 19 (Jensen's Inequality). Let cp : lR ---. lR be convex, and let X and cp(X) be integrable random variables. For any a-algebra Q, cpoE{XIQ}:S E{cp(X)IQ}. Corollary 1. Let X be a martingale, and let cp be convex such that cp(Xt ) is integrable, 0 :S t < 00. Then cp(X) is a submartingale. In particular, if M is a martingale, then IMI is a submartingale. Corollary 2. Let X be a submartingale and let cp be convex, non-decreasing, and such that cp(Xt)O$;t 12 I Preliminaries Then UA are stopping times for all U ::::: s. Moreover since E{XUA } = 0 by hypothesis, for u ::::: s. Thus for A E F s and s < t, E{XtI A} = E{XsIA} = -E{Xoo1!l\A}' which implies E{XtIFs} = X s, and X is a martingale, 0 :S t :S 00. 0 Definition. A martingale X with Xo = 0 and E{Xn < 00 for each t > 0 is called a square integrable martingale. If E{X~,} < 00 as well, then X is called an L 2 martingale. Clearly, any L2 martingale is also a square integrable martingale. See also Sect. 3 of Chap. IV. 3 The Poisson Process and Brownian Motion The Poisson process and Brownian motion are the two fundamental examples in the theory of continuous time stochastic processes. The Poisson process is the simpler of the two, and we begin with it. We recall that we assume given a filtered probability space (fl, F, IF, P) satisfying the usual hypotheses. Let (Tn)n~o be a strictly increasing sequence of positive random variables. We always take To = 0 a.s. Recall that the indicator function l{t~Tn} is defined as { I, if t ::::: Tn(w),l{t~Tn} = 0, if t < Tn(w). Definition. The process N = (Nt)o~t~oo defined by Nt = L l{t~Tn} n~l with values in NU{oo} where N = {O, I, 2, ... } is called the counting process associated to the sequence (Tn)n>l' If we set T = sUPn Tn, then [Tn, (0) = {N::::: n} = {(t,w): Nt(w)::::: n} as well as [Tn, Tn+d = {N = n}, and [T,oo) = {N = oo}. 3 The Poisson Process and Brownian Motion 13 The random variable T is the explosion time of N. IfT = 00 a.s., then N is a counting process without explosions. For T = 00, note that for 0 ::; s < t < 00 we have Nt - N s = L l{s.t, for some constant A ::::: O. 3 N is continuous in probability means that for t > 0, limu--->t Nu = Nt where the limit is taken in probability. 14 I Preliminaries Since {Nt = O} = {Ns = O} n {Nt - N s = O} for 0 ::; s < t < 00 by the independence of the increments, P(Nt = 0) = P(Ns = O)P(Nt - N s = 0) = P(Ns = O)P(Nt- s = 0), by the stationarity of the increments. Let a(t) = P(Nt = 0). We have a(t) = a(s)a(t - s), for all 0 ::; s < t < 00. Since a(t) can be easily seen to be right continuous in t, we deduce that either a(t) = 0 for all t ::::: 0 or a(t) = e->.t for some A ::::: O. If a(t) = 0 it would follow that Nt(w) = 00 a.s. for all t which would con- tradict that N is a counting process. Note that limu->t P(INu - Ntl > c) = limu->t P(INu-tl > c) = limv->o P(Nv > c) = limv->o 1 - e->'v = 0; hence N is continuous in probability. Step 2. P(Nt ::::: 2) is o(t). (That is, limt->o iP(Nt ::::: 2) = 0.) Let (3(t) = P(Nt ::::: 2). Since the paths of N are non-decreasing, (3 is also non-decreasing. One readily checks that showing limt->o t{3(t) = 0 is equivalent to showing that limn->oo n{3(~) = O. Divide [0,1] into n subintervals of equal length, and let Sn denote the number of subintervals containing at least two arrivals. By the independence and stationarity of the increments Sn is the sum of n Li.d. zero-one valued random variables, and hence has a Binomial distribution (n,p), where p = (3(~). Therefore E{Sn} = np = n{3(~). Since N is a counting process, we know the arrival times are strictly increasing; that is, Tn < Tn+1 a.s. Since Sn ::; N 1, if E{N1 } < 00 we can use the Dominated Convergence Theorem to conclude limn->oo n{3(~) = limn->oo E{Sn} = O. (That E{N1 } < 00 is a consequence of Theorem 34, established in Sect. 4). Also note that E{Nd < 00 implies N 1 < 00 a.s. and hence there are no explosions before time 1. This implies for fixed w, for n sufficiently large no subinterval has more than one arrival (otherwise there would be an explosion). Hence, limn->oo Sn (w) = 0 a.s. Step 3. limt->o tP{Nt = I} = A. Since P{Nt = I} = 1- P{Nt = O} - P{Nt ::::: 2}, it follows that . 1 . l-e->.t+ o(t)hm-P{Nt =I}=hm =A.t->O t t->o t Step 4. Conclusion. We write cp(t) = E{aNt }, for 0 'S: a ::; 1. Then for 0 ::; s < t < 00, the independence and stationarity of the increments implies that cp(t + s) = cp(t)cp(s) which in turn implies that cp(t) = etl/;(a). But 3 The Poisson Process and Brownian Motion 15 00 cp(t) = Lo:np(Nt = n) n=O 00 = P(Nt = 0) + o:P(Nt = 1) + L o:nP(Nt = n), n=2 and 'ljJ(0:) = cp'(O), the derivative of cp at O. Therefore () 1. cp(t) - 1 1. {P(Nt = 0) - 1 o:P(Nt = 1) 1 ()}'ljJ 0: = 1m = 1m + + -0 t t--->O t t--->O t t t = -A + Ao:. Therefore cp(t) = e->.t+>.at, hence Equating coefficients of the two infinite series yields for n = 0, 1,2, .... o Definition. The parameter A associated to a Poisson process by Theorem 23 is called the intensity, or arrival rate, of the process. Corollary. A Poisson process N with intensity A satisfies E{Nt} = At, Variance(Nt ) = Var(Nt ) = At. The proof is trivial and we omit it. There are other, equivalent definitions of the Poisson process. For example, a counting process N without explosion can be seen to be a Poisson process if for all s, t, 0 ::; s < t < 00, E{Nt } < 00 and Theorem 24. Let N be a Poisson process with intensity A. Then Nt - At and (Nt - At)2 - At are martingales. Proof Since At is non-random, the process Nt - At has mean zero and inde- pendent increments. Therefore E{Nt - At - (Ns - As)IFs } = E{Nt - At - (Ns - AS)} = 0, for 0 ::; s < t < 00. The analogous statement holds for (Nt - At)2 - At. 0 16 I Preliminaries Definition. Let H be a stochastic process. The natural filtration of H, denoted lFo = (:F?)o~t 3 The Poisson Process and Brownian Motion 17 The Brownian motion starts at x if P(Bo = x) = 1. The existence of Brownian motion is proved using a path-space construc- tion, together with Kolmogorov's Extension Theorem. It is simple to check that a Brownian motion is a martingale as long as E{/Bol} < 00. Therefore by Theorem 9 there exists a version which has right continuous paths, a.s. Actually, more is true. Theorem 26. Let B be a Brownian motion. Then there exists a modification of B which has continuous paths a.s. Theorem 26 is often proved in textbooks on probability theory (e.g., Breiman [23]). It can also be proved as an elementary consequence of Kol- mogorov's Lemma (Theorem 72 of Chap. IV). We will always assume that we are using the version of Brownian motion with continuous paths. We will also assume, unless stated otherwise, that C is the identity matrix. We then say that a Brownian motion B with continuous paths, with C = I the identity matrix, and with B o = x for Some x E jRn, is a standard Brownian motion. Note that for an jRn standard Brownian motion B, writing Bt = (Bl, ... , Bf), o :::: t < 00, then each Bi is an jR1 Brownian motion with continuous paths, and the Bi's are independent. We have already observed that a Brownian motion B with E{IBol} < 00 is a martingale. Another important elementary observation is the following. Theorem 27. Let B = (Bdooo 7rn B = t a.s., for a standard Brownian motion B. 18 I Preliminaries Proof. We first show convergence in mean square. We have 7rnB - t = 2::= {(Bti+l - BtJ2 - (tHl - tin tiE1fn where Yi are independent random variables with zero means. Therefore Next observe that (Bti+1 -Bd2/(tHl -ti) has the distribution of Z2, where Z is Gaussian with mean 0 and variance 1. Therefore which tends to 0 as n tends to 00. This establishes L 2 convergence (and hence convergence in probability as well). To obtain the a.s. convergence we use the Backwards Martingale Conver- gence Theorem (Theorem 14). Define Nn(w) = 'Tr-nB = L (Bti+ 1 (w) - Bt;(w))2, tiE1f_ n for n = -1, - 2, -3, . . .. Then it is straightforward (though notationally messy) to show that E{NnINn- 1 , N n- 2, ... } = Nn-l. Therefore Nn is a martingale relative to On = u{Nk , k :S n}, n = -1, -2, .... By Theorem 14 we deduce limn -+_ oo N n = limn -+oo 7rn B exists a.s., and since 7rn B converges to t in £2, we must have limn -+oo 7rn B = t a.s. as well. 0 Comments. As noted in the proofs, the proof is simple (and half as long) if we conclude only £2 convergence (and hence convergence in probability), instead of a.s. convergence. Also, we can avoid the use of the Backwards Martingale Convergence Theorem (Theorem 14) in the second half of the proof if we add the hypothesis that L n mesh(7rn ) < 00. The result then follows, after having proved the L 2 convergence, by using the Borel-Cantelli Lemma and Chebsyshev's inequality. Furthermore to conclude only £2 convergence we do not need the hypothesis that the sequence of partitions be refining. Theorem 28 can be used to prove that the paths of Brownian motion are of unbounded variation on compacts. It is this fact that is central to the difficulties in defining an integral with respect to Brownian motion (and martingales in general). 4 Levy Processes 19 Theorem 29. For almost all w, the sample paths t ~ Bt(w) of a standard Brownian motion B are of unbounded variation on any interval. Proof. Let A = [a, b] be an interval. The variation of paths of B is defined to be VA(W) = sup ~ IBti+1 - BtJ rrEP ti Err where P are all finite partitions of [a,b]. Suppose P(VA < 00) > O. Let 7fn be a sequence of refining partitions of [a, b] with limn mesh(7fn) = O. Then by Theorem 28 on {VA < oo}, b - a = lim '"' (Bti+ 1 - Bt;)2n~oo L.-t ti E1fn ::; lim sup IBti+l - B ti I 2::= IBti+ 1 - B ti I n---+oo tiE1rn tiE1r n ::; lim sup IBti+1 - BtJVA n---+oo tiE1Tn =0, since SUPtiErrn IBti+l - B ti I tends to 0 a.s. as mesh(7fn) tends to 0 by the a.s. uniform continuity of the paths on A. Since b- a ::; 0 is absurd, by Theorem 27 we conclude VA = 00 a.s. Since the null set can depend on the interval [a, bJ, we only consider intervals with rational endpoints a, b with a < b. Such a collection is countable, and since any interval (a,b) = U~=l[an,bn] with an, bn rational, we can omit the dependence of the null set on the interval. 0 We conclude this section by observing that not only are the increments of standard Brownian motion independent, they are also stationary. Thus Brow- nian motion is a Levy process (as is the Poisson process), and the theorems of Sect. 4 apply to'it. In particular, by Theorem 31 of Sect. 4, we can con- clude that the completed natural filtration of standard Brownian motion is right continuous. 4 Levy Processes The Levy processes, which include the Poisson process and Brownian motion as special cases, were the first class of stochastic processes to be studied in the modern spirit (by the French mathematician Paul Levy). They still pro- vide prototypic examples for Markov processes as well as for semimartingales. Most of the results of this section hold for JRn-valued processes; for notational simplicity, however, we will consider only JR-valued processes.4 Once again we recall that we are assuming given a filtered probability space (51, F, IF, P) satisfying the usual hypotheses. 4 lRn denotes n-dimensional Euclidean space. lR+ = [0,00) denotes the non-negative real numbers. 20 I Preliminaries Definition. An adapted process X = (Xdt>o with X o = 0 a.s. is a Levy process if (i) X has increments independent of the past; that is, Xt - Xs is independent of F s ' O:S s < t < 00; and (ii) X has stationary increments; that is, Xt - Xs has the same distribution as X t - s , 0 :s s < t < 00; and (iii) Xt is continuous in probability; that is, limt---+s Xt = Xs , where the limit is taken in probability. Note that it is not necessary to involve the filtration IF in the definition of a Levy process. Here is a (less general) alternative definition; to distinguish the two, we will call it an intrinsic Levy process. Definition. An process X = (Xdt;:::o with X o = 0 a.s. is an intrinsic Levy process if (i) X has independent increments; that is, Xt - Xs is independent of Xv - Xu if(u,v)n(s,t) =0; and (ii) X has stationary increments; that is, Xt - Xs has the same distribution as Xv - Xu if t - s = v - u > 0; and (iii) X t is continuous in probability. Of course, an intrinsic Levy process is a Levy process for its minimal (completed) filtration. Ifwe take the Fourier transform ofeach X t we get a function f (t, u) = ft (u) given by ft(u) = E{eiuX,}, where fo(u) 1, and ft+s(u) = ft(u)fs(u), and ft(u) -# 0 for every (t, u). Using the (right) continuity in probability we conclude ft(u) = exp{ -t'IjJ(u)}, for some continuous function 'IjJ(u) with 'IjJ(0) = O. (Bochner's Theorem can be used to show the converse. If 'IjJ is continuous, 'IjJ(0) = 0, and if for all t ~ 0, ft(u) = e-t'I/J(u) satisfies 2:i,j al5.jft(ui - Uj) ~ 0, for all finite (Ul' ... ,Un; 0'.1, ... ,an), then there exists a Levy process corresponding to f·) In particular it follows that if X is a Levy process then for each t > 0, Xt has an infinitely divisible distribution. Inversely it can be shown that for each infinitely divisible distribution jj there exists a Levy process X such that jj is the distribution of Xl. Theorem 30. Let X be a Levy process. There exists a unique modification Y of X which is cadlag and which is also a Levy process. iuX tProof· Let Mr = !,(u) . For each fixed u in iQ, the rationals in IR, the process (Mr)ost 4 Levy Processes 21 t f---> MtU(w) and t f---> eiuX,(w), with t E iQl+, are the restrictions to iQl+ of cadlag functions. Let A = {(w u) En x IR' eiuX,(w) t E Ifl\, . ,"'l+, is not the restriction of a d1dlag function}. One can check that A is a measurable set. Furthermore, we have seen that J lA(W, u)P(dw) = 0, each u E R By Fubini's Theorem JI: lA(w, u)duP(dw) = I:JlA(w, u)P(dw)du = 0, hence we conclude that for a.a. w the function t f---> eiuXt(w), t E Q+ is the restriction of a cadlag function for almost all u E R We can now conclude that the function t f---> Xt (w), t E iQl+, is the restriction of a cadlag function for every such w, with the help of the lemma that follows the proof of this theorem. Next set yt(w) = limsEiQI+,sLt Xs(w) for all w in the projection onto 51 of {51 x IR} \ A and yt = °on A, all t. Since Ft contains all the P-null sets of F and (Fdo 22 I Preliminaries Theorem 31. Let X be a Levy process and let Ot = nvN, where (n)o t and (UI, ... , un), we give the proof for n = 2 for nota- tional convenience. Therefore let z > v > t, and suppose given UI and Uz. We have E{ei(U 1 XV+U2 Xz)IOt+} = limE{ei(u 1 XV+U2 Xz)IOw} wlt eiU2Xz = limE{eiUIXV_(-)fz(U2)IQw} wlt fz Uz eiU2Xv = limE{eiUIXV_j( )fz(uz)IOw}, wlt v Uz using that M;:2 = e;v7u~) is a martingale. Combining terms the above becomes and the same martingale argument yields = lime i (Ul+U2)Xw fv-w(UI + u2)fz-v(U2) wlt = ei(Ul+U2)X, fv-t(UI + u2)fz-v(U2) = E{ei(UIXV+U2XZ)IOtl· It follows that E{eiEujXSjIOH} = E{eiEujXsj!Ot} for all (SI, ... ,Sn) and all (UI, ... ,Un ), whence E{ZIOt+} = E{ZIOtl for every bounded Z E VO 4 Levy Processes 23 Theorem 32. . Let X be a Levy process and let T be a stopping time. On the set {T < oo} the process Y = (yt)o:s:t- 24 I Preliminaries The process X is the Brownian motion B up to time T, and after time T it is the Brownian motion B "reflected" about the constant level z. Since Band - B have the same distribution, it follows from Theorem 32 that X is also a standard Brownian motion. Next we define R = inf{t > 0 : Xt = z}. Then clearly P(R :::: t; Xt < z - y) = P(T ~ t; B t < z - y), since (R, X) and (T, B) have the same distribution. However we also have that R = T identically, whence {R :::: t; Xt < z - y} = {T :::: t; Bt > z + y} by the construction of X. Therefore P(T :::: t; B t > z + y) = P(T:::: t; B t < z - y). The left side of (*) equals P(St ~ z; B t > z + y) = P(Bt > z + y), where the last equality is a consequence of the containment {St ~ z} ::> {Bt > z + y}. Also the right side of (*) equals P(St ~ z; B t < z - y). Combining these yields P(St ~ z; B t < z - y) = P(Bt > z + y), which is what waS to be proved. o We also have a reflection principle for Levy processes. See Exercises 30 and 31. Corollary. Let B = (Bdt?o be standard Brownian motion (Bo = 0 a.s.) and St = sUPo 0, P(St > z) = 2P(Bt > z). Proof Take y = 0 in Theorem 33. Then P(Bt < z; St ~ z) = P(Bt > z). Adding P(B t > z) to both sides and noting that {Bt > z} = {Bt > z} n {St ~ z} yields the result since P(Bt = z) = O. 0 4 Levy Processes 25 A Levy process is cadlag, and hence the only type of discontinuities it can have is jump discontinuities. Letting X t - = limsit X s , the left limit at t, we define ~Xt = Xt - X t-, the jump at t. If SUPt I~Xtl ::; C < 00 a.s., where C is a non-random constant, then we say that X has bounded jumps. Our next result states that a Levy process with bounded jumps has finite moments of all orders. This fact was used in Sect. 3 (Step 2 of the proof of Theorem 23) to show that E{NI} < 00 for a Poisson process N. Theorem 34. Let X be a Levy process with bounded jumps. Then E{IXtl n } < 00 for all n = 1, 2, 3, .... Proof Let C be a (non-random) bound for the jumps of X. Define the stop- ping times T I = inf{t : IXtl ~ C} Tn+1 = inf{t > Tn: IXt - XTn! ~ C}. Since the paths are right continuous, the stopping times (Tn)nZI form a strictly increasing sequence. Moreover I~XTI ::; C by hypothesis for any stopping time T. Therefore sups IX;n I ::; 2nC by recursion. Theorem 32 im- plies that Tn - Tn- I is independent of FTn _ 1 and also that the distribution of Tn - Tn- I is the same as that of T I . The above implies that for some a, 0 ::; a < 1. But also which implies that Xt has an exponential moment and hence moments of all orders. 0 We next turn Our attention to an analysis of the jumps of a Levy process. Let A be a Borel set in lR. bounded away from 0 (that is, 0 ¢:. if, where if is the closure of A). For a Levy process X we define the random variables Tl = inf{t > 0: ~Xt E A} T;;+I = inf{t > T;t : ~Xt E A}. 26 I Preliminaries Since X has d1dlag paths and 0 ¢:. iI, the reader can readily check that {TA2: t} E Ft+ = Ft and therefore each TA is a stopping time. Moreover 0 ¢:. iI and cadlag paths further imply Tj > 0 a.s. and that limn->oo TA = 00 a.s. We define 00 Nt = L lA(~Xs) = L l{TA~t} O 4 Levy Processes 27 Corollary. Let A be a Borel set of IR with 0 ¢: iI, and let f be Borel and finite on A. Then is a Levy process. For a given set A (as always, 0 ¢: iI), we defined the associated jump process to be Jt = ~ ~XsIA(~Xs)' O 28 I Preliminaries and also Proof· First let f = Lj aj 1Aj' a simple function. Then E{L ajNtAj } = L ajE{N/j } j = t L ajll(Aj ), j since N/j is a Poisson process with parameter lI(A j ). The first equality follows easily. For the second equality, let Mf = Nt' -tll(Ai ). The Mf are LP martingales, all p ~ 1, by the proof of Theorem 34. Moreover, E{Mf} = O. Suppose Ai, Aj are disjoint. We have E{MiMi} = E{i)ML+l - MiJ 2:)Mit+1 - Mit)} k i for any partition 0 = to < tl < ... < tn = t. Using the martingale property we have E{MfMl} = E{L(ML+l - ML)(Mlk+l - M1J}. k Using the inequality labl ::; a2 + b2 , we have 2:)ML+l - MiJ(ML+l - MiJ ::; I)ML+l - MfJ2 + 2:)ML+l - MiJ 2. k k k However Lk(Mlk+1 - M1J2 ::; (Nt')2 + 1I(Ai )2t 2; therefore the sums are dominated by an integrable random variable. Since Ml and Ml have paths of finite variation on [O,t] it is easy to deduce that if we take a sequence (1fn )n21 of partitions where the mesh tends to 0 we have nl!..~ L (ML+l - Mik)(ML+l - MiJ = L ~M;~M1- tk,tk+lE7rn O 4 Levy Processes 29 Remark. The first statement in Theorem 38 remains true if f1A E LI(dl/). See Exercise 28. Corollary. Let f : JR --* JR be bounded and vanish in a neighborhood of O. Then E{ L f(~Xs)} = t100 f(x)l/(dx). O 30 I Preliminaries which in turn implies, because of the independence and stationarity of the increments that E{ei(UtJt1+U2(Jt12-Jt1)+'-+Un(Jt1n -Jt1n _ 1 »ei(v1ff1+"-+Vn(J;n -J;n_l»} = E{e iUIJt\ +i ~j=2 Uj (Jlj -Jt~_l)}E{eiVrJ;l +i ~j=2 Vj (J;j -J;j_l )}. This is enough to give independence. o The preceding results combine to yield the following useful theorem, which is one of the fundamental results about Levy processes. Theorem 40. Let X be a Levy process. Then X t = yt + Zt, where Y, Z are Levy processes, Y is a martingale with bounded jumps, yt E LP for all p ::::: 1 and Z has paths of finite variation on compacts. Proof. Let Jt = Lo 4 Levy Processes 31 For this proof we take a = 1. Let Ak = {k~l < Ixl ::; k}. Then MAk are pairwise independent Levy processes and martingales (Theorem 39). Set M n = L:~=l M A k • Then the martingales Z - Mn and Mn are independent by an argument similar to the one in the proof of Theorem 39. Moreover Var(Zt) = Var(Zt - Mr) + Var(Mr) where Var(X) denotes the variance of a random variable X. Therefore Var(Mt') ::; Var(Zt) < 00 for all n. We deduce that Mr is Cauchy in L2 and hence converges in L2 as n tends to 00 to a martingale zf, and Z - Mn also converges to a martingale ZC. Using Doob's maximal quadratic inequality (Theorem 20), we can find a subsequence con- verging a.s., uniformly in t on compacts, which permits the conclusion that zc has continuous paths. The independence of Zd and ZC follows from the independence of Mn and Z - Mn, for every n. 0 Note that a consequence of the convergence of Mt' to Zf in £2 in the proof of Theorem 41 is that the integral Ir-l,O)U(O,l] x 2 11(dx) is finite. Note that this improves a bit on the conclusion in Theorem 38. We recall that for a set A, 0 ~ iI, the process Nt = fA Nt(·,dx) is a Poisson process with parameter lI(A), and thus Nt - tll(A) is a martingale. Definition. Let N be a Poisson process with parameter A. Then Nt - At is called a compensated Poisson process. Theorem 41 can be interpreted as saying that a Levy process with bounded jumps decomposes into the sum of a continuous martingale Levy process and a martingale which is a mixture of compensated Poisson processes. It is not hard to show that E{eiuZ~} = e-ta2u2 /2, which implies that Zc must be a Brownian motion. The full decomposition theorem then follows easily. We state it here without proof (consult Bertoin [12]' Bretagnolle [25], Feller [72], or Jacod-Shiryaev [110] for a proof). Theorem 42 (Levy Decomposition Theorem). Let X be a Levy process. Then X has a decomposition X t = B t + r x(Nt (-, dx) - tll(dx)) J{lxl 32 I Preliminaries Theorem 43. Let X be a Levy process with Levy measure v. Then where 'ljJ(u)=1J2u2_iau+ r (l-e iUX )v(dx)+ r (l-eiux +iux)v(dx). 2 J{lxl?l} J{lxl 4 Levy Processes 33 Af(x) = lim Ptf(x) - f(x) t---.O t n [Pf = - cry f(x), a) - L (J"jk ax ax (x) j,k=1 J k +! f(x + y) - f(x) - (\7 f(x), T](y))//(dy). Note further that if f is constant in a neighborhood of x, then lim Ptf(x) - f(x) = !f(x + y) - f(x)//(dy). t---.O t We next consider some examples of Levy processes. For the examples we take n=l. Example. If a Levy process X has continuous paths, then // is identically 0, and X must be Brownian motion with drift. That is, X t = (J"Bt + at, for constants a, (J", are the only Levy processes with continuous paths. Example. Assume that // is a finite measure. That is, //(lRn ) < 00, and that (J"2 = O. Then X is a compound Poisson process with jump arrival intensity>. = //(lRn ). That is, let Nt = L::l1{t2Ti} be a Poisson process with arrival intensity A. Let Zt = L::1 Ui 1{t2Ti } where the sequence (Uj )j21 are i.i.d., independent of the sequence (Tj )j21, and .qu1) = t//. Then one easily verifies E{eiuZt } = e-t,p(u), where 'ljJ(u) = f(l- eiUX)//(dx). Example. Let //(dx) = L:r=1 akc/h (dx), where c{3k (dx) denotes the positive point mass measure at f3k E lR of size one. Assume L:~1 f3~ak < 00 and (J"2 = O. Let N k be a sequence of independent Poisson processes with parameters ak. Then X t = L:r=1 f3k(Ntk - akt) has // as its Levy measure. Example. Let X be a Levy process which is a Gamma process. That is, .c(Xt ) = r(a, (3), with f3 2: 0 and a = at. More precisely, the density for X t is ft(x) = rl::)xta - 1e-{3x1{x>O}. In this case (J"2 = 0, and taking a = f3 = 1 gives (/Jt(u) = E{eiuZt } = (1 - iu)-t. (See, for example, page 43 of Jacod- Protter [109] or p. 73 of Bertoin [12].) Then (/Jt(u) = e-t1n(l-iu) = et,p(u). But 'ljJ'(u) = i 1. = i(h(u) = iE{eiuZ1} = i roo eiuxe-xdx.(1 - zu) io Integrating yields 100 eiux - 1'ljJ(u) = e-xdxo x which implies, by the Levy-Khintchine formula, that //(dx) = ~e-x1{x>o}dx. Example. Our last example is that of stable processes. We begin by recall- ing what a stable distribution is. Extending the notion of the Central Limit Theorem, consider the normalized sums 34 I Preliminaries S - L:~=1 Xi - (3 0n - n, 'Yn> . 'Yn We say that a random variable X has a stable law if it is the limit in distribution as n -+ 00 of sums of the above form. One can then prove (see, for example, Breiman [23, page 200]) that either X has a normal distribution, or there exists a number 0:,0 < 0: < 2, called the index of the stable law, and also constants m1 2: 0, m2 2: 0, and 8 such that E{eiuX } = e'l/J(u) and . 100 iux iux 1 'ljJ(u) = zu8 + m1 (e -1- --2)~+dx o l+x x Q 10 . 1iux zux+m2 (e -1- --2)-111+ dx.-00 1 + x X Q One says that a stable distribution is symmetric if m1 = m2. By analogy we define a symmetric stable process (in the case n = 1) to be a Levy process where the Levy measure has the form lI(dx) = ,x1d+o 5 Why the Usual Hypotheses? 35 Let X be a cfldlag process and let X be Markov with respect to IF. Since X is Markov, we can associate with it a Markov transition function on (]Rd, B), where B denotes the Borel sets of ]Rd. This is a family (Ps,t)s,tEIR+ of Markov kernels on (]Rd, B) such that for each triple (s, t, u) with s < t < u. We then have the fundamental rela- tionship E{f(Xt)IFs} = Ps,t(Xs, j) a.s. Note that, by using indicator functions such as lA = f, we have Also note that the above implies the equality which, of course, is weaker. That is, the converse implication is not true in general. The most important special case is Markov processes which are time homogeneous. In this case we have a transition semigroup Pt given by for s ::; t. Definition. Let Co denote continuous functions vanishing at infinity. A Markov process is a Feller process if its transition semigroup (Ptk~o has the following two properties. (a) For each f E Co and each t > 0, Pd E Co· (b) For each f E Co and each x E ]Rd, Pd(x) -+ f(x) as t -+ O. Examples of Feller processes include Brownian motion and the Poisson process. More generally, one can show that diffusions5 and all Levy processes are Feller processes. Theorem 46. Let X be a cadlag Feller process for a filtration IF. Then it is also Markov with respect to the filtration IF+ = (FH)o:o;t:o;oo. Proof. Let t> 0 be fixed, and assume f E Co. Let yt = f(Xt ). For 0 ::; s ::; t, let Then Y is a martingale for 0 ::; s ::; t, and it is cadlag, because Pd is continuous, and X is cadlag. Therefore, Y is a martingale with respect to IF+ too, which means that E{f(Xt)!Fs+} = ~, from which we deduce that E{f(Xt)IFs+} = E{f(Xt)IXs}, and we are done. 0 5 This depends, of course, on how one defines a diffusion; the definition is not yet standardized in the literature. 36 I Preliminaries We now define pI' as a probability measure such that X is Markov with respect to pI' and also £(Xo) = J.L under PI', where £(Xo) denotes the distri- bution, or "law," of X o. Let IFJ1. = (Ff)o::;t::;oo be the filtration where Ff de- notes the pI' completion of Ff, Ff = o-{Xs : s ::; t}, and.ro = o-{Xs : s ~ O}. Also, let F t = n{IIJ1.II=l} Ft Theorem 47. If X is Markov with respect to pI', or alternatively Markov with respect to pI' for all J.L with IIJ.LII = 1, then IFJ1. and IF are, respectively, right continuous. Proof. We prove the result for IFJ1., from which the result for IF follows. Let 1i denote all h E bFJ1. such that E{h/Ff} = E{h/Ff+}, PJ1.-a.s. Then 1i ::) {H = f(Xs ), f EbB}, which is obvious for s ::; t, and for s > t it is just Theorem 46. Then"H contains all H ofthe form H = TI~=l fi(Xs.), for any f EbB. Using the Monotone Class Theorem, 1i ::) bfO. By completion, 1i ::) bFJ1.. Therefore if h E Ff+, we have E{hIFf} = E{hIFf+} = h, PJ1._ a.s. 0 The property that the natural filtration of a Feller process is right con- tinuous when completed is not restricted to Feller processes. Indeed, it can also be shown to be true for more general classes of processes, such as Hunt processes6 . However it is simply two elementary properties which yield the right continuity of the filtration in a general setting. Definition. Let X be a dtdlag Markov process for a filtration IF, which we take to be completed and therefore can assume to be right continuous, in view of Theorem 47. Let T be a stopping time. We say that X verifies the strong Markov property at T if We say that X is strong Markov if the above holds for every stopping time T. Suppose now that the semigroup (Pt ) is Borel. That is, if fEB, then Ptf E B as well. Suppose that for every probability measure J.L on (lRd ,B), there exists a probability space (D, 9, Q) and a cadlag Markov process (1-t)t20 defined on this space with £(Xo) = J.L. Suppose further that every such dtdlag Markov process Y is strong Markov. Then the filtrations IFJ1. = (Ff)o::;t::;oo and IF = (Ft)o::;t::;oo can be shown to be right continuous in a manner analogous to the previous proofs. 6 See, for example, Sharpe [215, page 219] for a definition of a Hunt process. 6 Local Martingales 37 6 Local Martingales Recall that we assume as given a filtered probability space (Q, F, IF, P) satis- fying the usual hypotheses. For a process X and a stopping time T we further recall that X T denotes the stopped process Definition. An adapted, cadlag process X is a local martingale if there exists a sequence of increasing stopping times, Tn' with limn --+oo Tn = 00 a.s. such that XU\Tn l{Tn >O} is a uniformly integrable martingale for each n. Such a sequence (Tn) of stopping times is called a fundamental sequence. Example. Clearly any cadlag martingale is a local martingale (take Tn == n). Example. We give an example of a local martingale which is not a martingale. Let (Bt)oO} is to relax the integrability condition on Xo. This is useful, for example, in the consideration of stochastic integral equations with a non-integrable initial condition. Definition. A stopping time T reduces a process M if M T is a uniformly integrable martingale. Theorem 48. Let M, N be local martingales and let Sand T be stopping times. (a) If T reduces M and S ::::; T a. s., then S reduces M. (b) The sum M + N is also a local martingale. (c) If S, T both reduce M, then S V T also reduces M. (d) The processes M T , M T 1{T>O} are local martingales. (e) Let X be a cadlag process and let Tn be a sequence of stopping times increasing to 00 a.s. such that XTn l{Tn >o} is a local martingale for each n. Then X is a local martingale. Proof. (a) follows from the Optional Sampling Theorem (Theorem 16) and Theorem 13. For (b), if (Sn)n2I, (Tn)n2I are fundamental sequences for M 38 I Preliminaries and N, respectively, then Sn 1\ Tn is a fundamental sequence for M + N. For (c), let Xt = M t - Mo. Then XSVT = XS + XT - XSI\T is a uniformly integrable martingale. But which is in L l , therefore XsvT +Mol{sVT>O} = M SVT 1{SVT>O} is a uniformly integrable martingale. The proof of (d) consists of the observation that if (Tn)n>l is a fundamental sequence for M, then it is also one for M T and M T 1{-;',>o} by the Optional Sampling Theorem (Theorem 16). For (e), we have XTnI{Tn>o} = Mn is a local martingale for each n. For fixed n we know there exists a fundamental sequence Un,k increasing to 00 a.s. as k tends to 00. For each n, choose k = ken) such that p(un,k(n) < Tn 1\ n) < 2-n . Then limn un,k(n) = 00 a.s., and un,k(n) 1\ Tn reduces X for each n. We take R m = max(Ul,k(l) 1\ T l , ... , um,k(m) 1\ Tm ), and each R m reduces X by (c), the R m are increasing, and lim Rm = 00 a.s. Therefore X is a local martingale. 0 Corollary. Local martingales form a vector space. We will often need to know that a reduced local martingale, M T , is in LP and not simply uniformly integrable. Definition. Let X be a stochastic process. A property Jr is said to hold locally if there exists a sequence of stopping times (Tn)n>l increasing to 00 a.s. such that XTn I{Tn>O} has property Jr, each n :::: 1. - We see from Theorem 48(e) that a process which is locally a local mar- tingale is also a local martingale. Other examples of local properties that arise frequently are locally bounded and locally square integrable. Theorems 49 and 50 are consequences of Theorem 48(e). Theorem 49. Let X be a process which is locally a square integrable martin- gale. Then X is a local martingale. The next theorem shows that the traditional "uniform integrability" as- sumption in the definition of local martingale is not really necessary. Theorem 50. Let M be adapted, cadldg and let (Tn)n>l be a sequence of stopping times increasing to 00 a.s. If MTn I{Tn>O} is a ~artingale for each n, then M is a local martingale. It is often of interest to determine when a local martingale is actually a martingale. A simple condition involves the maximal function. Recall that X; = sUPs: 7 Stieltjes Integration and Change of Variables 39 Proof. Let (Tn)n2:1 be a fundamental sequence of stopping times for X. If s :S t, then E{XU\TnlFs } = XSI\Tn • The Dominated Convergence Theorem yields E{XtIFs } = X s ' If E{X*} < 00, since each IXtl :S X*, it follows that (Xt )t2:o is uniformly integrable. 0 Note that in particular a bounded local martingale is a uniformly inte- grable martingale. Other sufficient conditions for a local martingale to be a martingale are given in Corollaries 3 and 4 to Theorem 27 in Chap. II, and Kazamaki's and Novikov's criteria (Theorems 40 and 41 of Chap. III) establish the result for the important special cases of continuous local martingales. 7 Stieltjes Integration and Change of Variables Stochastic integration with respect to semimartingales can be thought of as an extension of path-by-path Stieltjes integration. We present here the es- sential elementary ideas of Stieltjes integration appropriate to our interests. We assume the reader is familiar with the Lebesgue theory of measure and integration on JR+. Definition. Let A = (At )t>o be a cadlag process. A is an increasing pro- cess if the paths of A : t ~ At(w) are non-decreasing for almost all w. A is called a finite variation process (FV) if almost all of the paths of A are of finite variation on each compact interval of JR+. Let A be an increasing process. Fix an w such that t f---t At(w) is right continuous and non-decreasing. This function induces a measure J.tA(W, ds) on JR+. If f is a bounded, Borel function on JR+, then J~ f(s)J.tA(w, ds) is well-defined for each t > O. We denote this integral by J; f(s)dAs(w). If Fs = F(s,w) is bounded and jointly measurable, we can define, w-by-w, the integral I(t,w) = J~ F(s,w)dAs(w). I is right continuous in t and jointly measurable. Proceeding analogously for A an FV process (except that the induced measure J.tA(W, ds) can have negative measure; that is, it is a signed measure), we can define a jointly measurable integral I(t,w) = it F(s,w)dAs(w) for F bounded and jointly measurable. Let A be an FV process. We define 2n IAlt = sup'"'" IA tk - At(k;;:l) I·~ 2""t" 2 n2: 1k =1 Then IAlt < 00 a.s., and it is an increasing process. 40 I Preliminaries Definition. For A an FV process, the total variation process, IAI = (/Altk~o, is the increasing process defined in (*) above. Notation. Let A be an FV process and let F be a jointly measurable process such that J; F(s,w)dAs(w) exists and is finite for all t > 0, a.s. We let (F· A)t(w) = it F(s,w)dAs(w) and we write F· A to denote the process F· A = (F . Atk~o. We will also write J; FsldAs/ for (F ·IA/k The next result is an absolute continuity result for Stieltjes integrals. Theorem 52. Let A, C be adapted, strictly increasing processes such that C - A is also an increasing process. Then there exists a jointly measurable, adapted process H (defined on (0,00)) such that 0 ::::; H ::::; 1 and A=H,C or equivalently At = it HsdCs. Proof If J.L and v are two Borel measures on lR+ with J.L« v, then we can set { tt«(s,t] o:(s, t) = v«s,t' 0, if v((s, t]) > 0, otherwise. Defining hand k by h(t) = liminfsTt o:(s, t), k(t) = limsuPtls o:(s, t), then h and k are both Borel measurable, and moreover they are each versions of the Radon-Nikodym derivative. That is, dJ.L = h dv, dJ.L = kdv, To complete the proof it suffices to show that we can follow the above proce- dure in a (t, w) measurable fashion. With the convention ~ = 0, it suffices to define ( ) 1" f (A(t,w) - A(rt,w))H t w = Imlll . , rp,rEQ+ (C(t,w) - C(rt,w))' such an H is clearly adapted since both A and Care. o Corollary. Let A be an FV process. There exists a jointly measurable, adapted process H, -1 ::::; H ::::; 1, such that or equivalently IAI=H·A and A=H·/AI and 7 Stieltjes Integration and Change of Variables 41 Proof We define At = ~(IAlt+At) and At = ~(IAlt-At). ThenA+ and A- are both increasing processes, and IAI- A+ and IAI- A- are also increasing processes. By Theorem 52 there exist processes H+ and H- such that At = J; H:ldAsl, At = J; H; IdAsl· It then follows that At = At -At = J;(H:- H;)ldAsl. Let Ht == Ht - Ht- and suppose H+ and H- are defined as in the proof of Theorem 52. Except for a P-null set, for a given w it is clear that IHs(w)1 = 1 dAs(w) almost all s. Considering H . A, we have ItHsdAs = it Hsd(ls HuidAuD = it HsHsldAsl = it IldAs l = IAlt. This completes the proof. o When the integrand process H has continuous paths, the Stieltjes integral J; HsdAs is also known as the Riemann-Stieltjes integral (for fixed w). In this case we can define the integral as the limit of approximating sums. Such a result is proved in elementary textbooks on real analysis (e.g., Protter- Morrey [195, pages 316, 317]). Theorem 53. Let A be an FV process and let H be a jointly measurable process such that a.s. s f---> H(s,w) is continuous. Let 1fn be a sequence of finite random partitions of [0, t] with limn --+ oo mesh(1fn ) = o. Then for Tk :S Sk :S Tk+l' We next prove a change of variables formula when the FV process is continuous. Ito's formula (Theorem 32 of Chap. II) is a generalization of this result. Theorem 54 (Change of Variables). Let A be an FV process with con- tinuous paths, and let f be such that its derivative l' exists and is continuous. Then (f(At )}t;2:o is an FV process and f(At ) - f(Ao) = I t f'(As)dA s. Proof For fixed w, the function s f---+ f'(As(w)) is continuous on [0, t] and hence bounded. Therefore the integral J; f'(As)dAs exists. Fix t and let 1fn be a sequence of partitions of [0, t] with limn --+ oo mesh(1fn ) = o. Then f(At ) - f(Ao) = L {f(Atk+l) - f(Atk )} tk,tk+l E7fn = L f'(ASk)(Atk+l - A tk ), k 42 I Preliminaries by the Mean Value Theorem, for some Sk, tk ~ Sk ~ tk+1. The result now follows by taking limits and Theorem 53. 0 Comment. We will see in Chap. II that the sums L f(Atk )(Atk+l - Atk ) tkE7fn [O,t] converge in probability to I t f(As-)dA s for a continuous function f and an FV process A. This leads to the more general change of variables formula, valid for any FV process A, and f E C 1 , namely f(At ) - f(Ao) = it f'(As-)dAs + l: {f(As) - f(A s-) - f'(As-)~As}. o O 8 NaIve Stochastic Integration Is Impossible 43 E{It - IsIFs} = E{l t HudMulFs} = E{ lim ~ Htk(Mtk+l - Mtk)IFs} n----i'OO L-, tk,tk+l E1rn = J~~ L E{E{Htk(Mtk+l - Mtk)IFtk}IFs} tk,tk+l E1rn =0. The interchange of limits can be justified by the Dominated Convergence The- orem. We conclude that the integral process I is a martingale. This fact, that the stochastic Stieltjes integral of an adapted, bounded, continuous process with respect to a martingale is again a martingale, is true in much greater generality. We shall treat this systematically in Chap. II. 8 NaIve Stochastic Integration Is Impossible In Sect. 7 we saw that for an FV process A, and a continuous integrand process H, we could express the integral J; HsdAs as the limit of sums (The- orem 53). The Brownian motion process, B, however, has paths of infinite variation on compacts. In this section we demonstrate, with the aid of the Banach-Steinhaus Theorem7 , some of the difficulties that are inherent in try- ing to extend the notion of Stieltjes integration to processes that have paths of unbounded variation, such as Brownian motion. For the reader's convenience we recall the Banach-Steinhaus Theorem. Theorem 55. Let X be a Banach space and let Y be a normed linear space. Let {To,} be a family of bounded linear opemtors from X into Y. If for each x E X the set {T",x} is bounded, then the set {liT", II} is bounded. Let us put aside stochastic processes for the moment. Let x(t) be a right continuous function on [0,1], and let 1fn be a refining sequence of dyadic rational partitions of [0,1] with limn -.+oo mesh(1fn ) = O. We ask the question "What conditions on x are needed so that the sums converge to a finite limit as n -+ 00 for all continuous functions h?" From Theorem 53 of Sect. 7 we know that x of finite variation is sufficient. However, it is also necessary. Theorem 56. If the sums Sn of (*) converge to a limit for every continuous function h then x is of finite variation. 7 The Banach-Steinhaus Theorem is also known as the Principle of Uniform Bound- edness. 44 I Preliminaries Proof. Let X be the Banach space of continuous functions equipped with the supremum norm. Let Y be JR, equipped with absolute value as the norm. For hEX, let For each fixed n it is simple to construct an h in X such that h(tk) sign{x(tk+1) - X(tk)}, and Ilhll = 1. For such an h we have Tn(h) = I: IX(tk+1) - x(tk)l· tk ,tk+l E7fn Therefore IITnl1 ~ L IX(tk+1) - x(tk)l, tk ,tk+1 E7fn each n, and sUPn IITnl1 2: total variation of x. On the other hand for each hEX we have limn_oo Tn(h) exists and therefore sUPn IITn(h)11 < 00. The Banach-Steinhaus Theorem then implies that sUPn IITnl1 < 00, hence the total variation of x is finite. 0 Returning to stochastic processes, we might hope to circumvent the lim- itations imposed by Theorem 56 by appealing to convergence in probability. That is, if X(s,w) is right continuous (or even continuous), can we have the sums I: Htk ( X t k+l - X tk ) tk ,tk+l E7fn [0,1] converging to a limit in probability for every continuous process H? Unfortu- nately the answer is that X must still have paths of finite variation, a.s. The reason is that one can make the procedure used in the proof of Theorem 56 measurable in w, and hence a subsequence of the sums in (**) can be made to converge a.s. to +00 on the set where X is not of finite variation. If this set has positive probability, the sums cannot converge in probability. The preceding discussion makes it appear impossible to develop a coherent notion of a stochastic integral J~ HsdXs when X is a process with paths of infinite variation on compacts; for example a Brownian motion. Nevertheless this is precisely what we will do in Chap. II. Bibliographic Notes The basic definitions and notation presented here have become fundamental to the modern study of stochastic processes, and they can be found many places, such as Dellacherie-Meyer [45], Doob [55], and Jacod-Shiryaev [110]. Theorem 3 is true in much greater generality. For example the hitting time of a Borel set is a stopping time. This result is very difficult, and proofs can be found in Dellacherie [41J or [42]. Exercises for Chapter I 45 The resume of martingale theory consists of standard theorems. The reader does not need results from martingale theory beyond what is presented here. Those proofs not given can be found in many places, for example Breiman [23], Dellacherie-Meyer [46], or Ethier-Kurtz [71]. The Poisson process and Brownian motion are the two most important stochastic processes for the theory of stochastic integration. Our treatment of the Poisson process follows Qinlar [33]. Theorem 25 is in Bremaud [24], and is due to J. de Sam Lazaro. The facts about Brownian motion needed for the theory of stochastic integration are the only ones presented here. A good source for more detailed information on Brownian motion is Revuz-Yor [208], or Hida [88]. Levy processes (processes with stationary and independent increments) are a crucial source of examples for the theory of semimartingales and stochastic integrals. Indeed in large part the theory is abstracted from the properties of these processes. There do not seem to be many presentations of Levy processes concerned with their properties which are relevant to stochastic integration beyond that of Jacod-Shiryaev [110]. However, one can consult the books of Bertoin [12], Rogers-Williams [210], and Revuz-Yor [208]. Our approach is inspired by Bretagnolle [25]. Local martingales were first proposed by K. Ito and S. Watanabe [102] in order to generalize the Doob-Meyer decomposition. Standard Stieltjes integration applied to finite variation stochastic processes was not well known before the fundamental work of Meyer [171]. Finally, the idea of using the Banach-Steinhaus Theorem to show that nalve stochastic integration is impossible is due to Meyer [178]. Exercises for Chapter I For all of these exercises, we assume as given a filtered probability space (Q, F, IF, P) satisfying the usual hypotheses. Exercise 1. Let 5, T be stopping times, 5 ~ T a.s. Show Fs eFT. Exercise 2. Give an example where 5, T are stopping times, 5 ~ T, but l' - 5 is not a stopping time. Exercise 3. Let (Tn)n~l be a sequence of stopping times. Show that sUPn Tn, infn Tn, lim sUPn--+oo Tn, and lim infn--+oo Tn are all stopping times. Exercise 4. Suppose (Tn)n>l is a sequence of stopping times decreasing to a random time T. Show that T is a stopping time, and moreover that FT = nn FTn • Exercise 5. Let p > 1 and let Mn be a sequence of continuous martingales (that is, martingales whose paths are continuous, a.s.) with M;;" E LP, each n. Suppose M;;" ~ X in LP and let Mt = E{XIFt}. (a) Show Mt E LP, all t ~ O. 46 Exercises for Chapter I (b) Show M is a continuous martingale. Exercise 6. Let N = (Ntk::o be Poisson with intensity A. Suppose At is an . _ 2(At)At e-At mteger. Show that E{INt - Atl} - (At-I)! . Exercise 7. Let N = (Nt)t>o be Poisson with intensity A. Show that N is continuous in L 2 (and hence-in probability), but of course N does not have continuous sample paths. Exercise 8. Let Bt = (Bl, B; , Bf) be three dimensional Brownian motion (that is, the B i are i.i.d. one dimensional Brownian motions, 1 :s:; i :s:; 3). Let To: = sUPt{IIBtll :s:; o:}. The time To: is known as a last exit time. Show that if 0: > 0, then To: is not a stopping time for the natural filtration of B. Exercise 9. Let (Ntk;::o be a sequence of Li.d. Poisson processes with pa- rameter A = 1. Let Mf = f(Nf - t), and let M = 0:=:1 Mfk;::o. (a) Show that M is well-defined (Le., that the series converges) in the L 2 sense. *(b) Show that for any t > 0, Ls O. Exercise 11. Show that a compound Poisson process is a Levy process. Exercise 12. Let Z be a compound Poisson process, with E{IU11} < 00. Show that Zt - E{UdAt is a martingale. Exercise 13. Let Z = (Zt)t>o be a Levy process which is a martingale, but Z has no Brownian component (that is, the constant a 2 = 0 in the Levy- Khintchine formula), and a finite Levy measure 1/. Let A = I/(R.). Show that Z is a compensated compound Poisson process with arrival intensity A and Li.d. jumps with distribution J.L = *1/. *Exercise 14. Let 0 < T1 < T2 < '" < Tn < ... be any increasing sequence of finite-valued stopping times, increasing to 00. Let Nt = L:l1{t2T;}, and let (Ui )i21 be independent, and also the (Ui )i21 are independent of the pro- cess N. Show that if SUPi E{IUil} < 00, all i, and E{Ud = 0, all i, then Zt = L:l Ui l{t2T;} is a martingale. (Note: We are not assuming that N is a Poisson process here. Furthermore, the random variables (Ui )i21, while independent, are not necessarily identically distributed.) Exercise 15. Let Z be a Levy process which is a martingale, but with no Brownian component and with a Levy measure of the form Exercises for Chapter I 47 00 v(dx) = 2: akclh (dx), k=1 where c,13k (dx) denotes point mass at (3k E lR of size one, with 2:%"=1 (3~ak < 00. Show that Z is of the form 00 Zt = 2: (3k(Ntk - akt ) k=1 where (Ntk)t>o is an independent sequence of Poisson processes with param- eters (akk:~;. Verify that Z is an L2 martingale. Exercise 16. Let B = (Bt)O~t~1 be one dimensional Brownian motion on [0,1]. Let W t = B 1- t - B1, 0 ::; t ::; 1. Show that W has the Same distribution as does B, and that W is a Brownian motion for 0 ::; t ::; 1 for its natural filtration. Exercise 17. Let X be any process with dl,dlag paths. Set c > O. (a) Show that on [0, t] there are only a finite number of jumps larger than c. That is, if Nt = 2:s O. - (b) Conclude that X can have only a countable number of jumps on [0, t]. Exercise 18. Let Z be a Levy process, and let c > O. Let ~Zs = Zs - Zs-, and set Ji = 2:s 48 Exercises for Chapter I (b) Let Y(w) = (1 - w)-a for some a, 0 < a < 1/2. Show that Y E £2 and also if Mt = E{YIFd, then 1 1 1 Mt(w) = (1 _ a) (1 _ t)a = 1 _ a Y(t), t < w < 1. (c) Show SUPt Mt = 12 a Y, w-by-w, and deduce 1 1II sup Mil£> = --IWII£> = --IIMooll£>· t I-a I-a Note that lima -+1/2(1 - a)-l = 2. Exercise 22. (This exercise gives an example of a martingale with each path bounded but which nevertheless is not locally bounded.) Let (n, F, JF, P) be as in Exercise 21. Let Y(w) = ./w, and Mt = E{YIFd· (a) Show that 1 2 Mt(w) = y'Wl(O,t j (w) + 1 + Jt 1(t,1)(w). (b) Let T be a stopping time, T not identically O. Show that T(w) 2:: w on some w-interval (0,10) with 10 > O. (c) Use (b) to show E{Mf.} 2:: J; ~ds = 00, and deduce that M cannot be locally an £2 martingale. (d) Conclude that M has bounded paths but that MT l{T>o} is unbounded for every stopping time T not identically O. *Exercise 23. Find an example of a local martingale M and finite stop- ping times S, T with S ::; T a.S. such that E{IMTIIFs} < 00, but Ms =I- E{MTIFs} a.s. (Hint: Begin by showing that the equality would imply that every positive local martingale is a martingale.) Exercise 24. (This exercise helps to clarify the difference between stopping time o--algebras of the two forms gT- and gT.) Let Z be a Levy process, and let gt = o-{ Zsi s ::; t} V N, where N are all the null sets in the given complete probability space (n, F, P). For a stopping time T define gT- = o-{A n (t < T); A E gdi at time 0, set go- = go. For a process X, let X T- = X t l{t Exercises for Chapter I 49 Exercise 25. Let Z be a Levy process (or any process which is continuous in probability). Show that the probability Z jumps at a given fixed time t is zero. Exercise 26. Use Exercises 24 and 25 to show that with the assumptions of Exercise 24, for any fixed time t, gt- = gt, giving a continuity property of the filtration for fixed times (since by Theorem 31 we always have gt+ = gt), but of course not for stopping times. Exercise 27. Let S, T be two stopping times with S ::::: T. Show fS- C fT-. Moreover if (Tn)n> 1 is any increasing sequence of stopping times with limn - HXl Tn = T, show that n Exercise 28. Prove the first equality of Theorem 38 when flA E L 1 (dv). **Exercise 29. Show that if Z is a Levy process and a local martingale, then Z is a martingale. That is, all Levy process local martingales are actually martingales. Exercise 30 (reflection principle for Levy processes). Let Z be a sym- metric Levy process with Zo = O. That is, Z and - Z have the same dis- tribution. Let St = sUPsS;t Zs. Show that P(St ::::: z; Zt < z - y) ::::: P(Zt > z + y). Exercise 31. Let Z be a symmetric Levy process with Zo = 0 as in Exer- cise 30. Show that P(St ::::: z) ::::: 2P(Zt ::::: z). II Semimartingales and Stochastic Integrals 1 Introduction to Semimartingales The purpose of the theory of stochastic integration is to give a reasonable meaning to the idea of a differential to as wide a class of stochastic processes as possible. We saw in Sect. 8 of Chap. I that using Stieltjes integration on a path-by-path basis excludes such fundamental processes as Brownian motion, and martingales in general. Markov processes also in general have paths of unbounded variation and are similarly excluded. Therefore we must find an approach more general than Stieltjes integration. We will define stochastic integrals first as a limit of sums. A priori this seems hopeless, since even by restricting our integrands to continuous pro- cesses we saw as a consequence of Theorem 56 of Chap. I that the differential must be of finite variation on compacts. However an analysis of the proof of Theorem 56 offers some hope. In order to construct a function h such that h(tk) = sign(x(tk+l) - X(tk)), we need to be able to "see" the trajectory of x on (tk, tk+l]. The idea of K. Ito was to restrict the integrands to those that could not see into the future increments, namely adapted processes. The foregoing considerations lead us to define the stochastic processes that will serve as differentials as those that are "good integrators" on an appropri- ate class of adapted processes. We will, as discussed in Chap. I, assume that we are given a filtered, complete probability space (n, F, JF, P) satisfying the usual hypotheses. Definition. A process H is said to be simple predictable if H has a rep- resentation n Ht = Hol{o}(t) + L Hi l(Ti,T;+l](t) i=l where 0 = Tl :s ... ::::: Tn +1 < 00 is a finite sequence of stopping times, Hi E FTi with IHil < 00 a.s., 0 ::::: i :s n. The collection of simple predictable processes is denoted S. 52 II Semimartingales and Stochastic Integrals Note that we can" take T 1 = To = 0 in the above definition, so there is no "gap" between To and T1" We can topologize S by uniform convergence in (t,w), and we denote S endowed with this topology by SUo We also write LO for the space of finite-valued random variables topologized by convergence in probability. Let X be a stochastic process. An operator, Ix, induced by X should have two fundamental properties to earn the name "integral." The operator I x should be linear, and it should satisfy some version of the Bounded Con- vergence Theorem. A particularly weak form of the Bounded Convergence Theorem is that the uniform convergence of processes Hn to H implies only the convergence in probability of Ix (Hn ) to Jx(H). Inspired by the above considerations, for a given process X we define a linear mapping Ix : S ----; LO by letting n Ix(H) = HoXo +I: Hi (XTi+1 - XTJ, i=l where H E S has the representation n Ht = H ol{o} +L H i1(Ti,Ti+1 ]. i=l Since this definition is a path-by-path definition for the step functions H(w), it does not depend on the choice of the representation of H in S. Definition. A process X is a total semimartingale if X is dtdlag, adapted, and Ix : Su ----; LO is continuous. Recall that for a process X and a stopping time T, the notation X T denotes the process (Xt/\T )t>o. Definition. A process X is called a semimartingale if, for each t E [0,00), xt is a total semimartingale. With our definition of a semimartingale, the second fundamental property we want (bounded convergence) holds. We postpone consideration of examples of semimartingales to Sect. 3. 2 Stability Properties of Semimartingales We state a sequence of theorems giving some of the stability results which are particularly simple. Theorem 1. The set of (total) semimartingales is a vector space. Proof This is immediate from the definition. o 2 Stability Properties of Semimartingales 53 Theorem 2. If Q is a probability and absolutely continuous with respect to P, then every (total) P semimartingale X is a (total) Q semimartingale. Proof. Convergence in P-probability implies convergence in Q-probability. Thus the theorem follows from the definition of X. 0 Theorem 3. Let (Pkk?l be a sequence of probabilities such that X is a Pk semimartingale for each k. Let R = L~=l AkPk, where Ak ::::: 0, each k, and L~=l Ak = 1. Then X is a semimartingale under R as well. Proof Suppose H n E S converges uniformly to H E S. Since X is a Pk semimartingale for all Pk, Ix(Hn) converges to Ix (H) in probability for every Pk. This then implies Ix (Hn) converges to Ix(H) under R. 0 Theorem 4 (Stricker's Theorem). Let X be a semimartingale for the filtmtion IF. Let G be a subfiltmtion of IF, such that X is adapted to the G filtmtion. Then X is a G semimartingale. Proof For a filtration JBI, let S(JBI) denote the simple predictable processes for the filtration JBI = (1tt k?o. In this case we have S(G) is contained in S(IF). The theorem is then an immediate consequence of the definition. 0 Theorem 4 shows that we can always shrink a filtration and preserve the property of being a semimartingale (as long as the process X is still adapted), since we are shrinking as well the possible integrands; this, in effect, makes it "easier" for the process X to be a semimartingale. Expanding the filtra- tion, therefore, should be--and is-a much more delicate issue. Expansion of filtrations is considered in much greater detail in Chap. VI. We present here an elementary but useful result. Recall that we are given a filtered space (n, F, IF, P) satisfying the usual hypotheses. Theorem 5 (Jacod's Countable Expansion). Let A be a collection of events in F such that if A o" A,a E A then Au n A,a = 0, (a =1= (3). Let 1tt be the filtmtion genemted by F t and A. Then every (IF, P) semimartingale is an (JBI, P) semimartingale also. Proof Let An EA. If P(An) = 0, then An and A~ are in Fo by hypothesis. We assume, therefore, that P(An ) > O. Note that there can be at most a countable number of An E A such that P(An) > o. If A = Un>l An is the union of all An E A with P(An) > 0, we can also add Ac to A without loss of generality. Thus we can assume that A is a countable partition of n with P(An ) > 0 for every An EA. Define a new probability Qn by Qn(-) = P(·IAn), for An fixed. Then Qn « P, and X is a (IF, Qn) semimartingale by Theorem 2. If we enlarge the filtration IF by all the F measurable events that have Qn- probability 0 or 1, we get a larger filtration Jfn = (:ltk~.o, and X is a (Jfn, Qn) semimartingale. Since Qn(Am ) = 0 or 1 for m =1= n, we have F t C 1tt c :It, for t ::::: 0, and for all n. By Stricker's Theorem (Theorem 4) we conclude that X is an (JBI, Qn) semimartingale. Finally, we have dP = Ln?l P(An)dQn, 54 II Semimartingales and Stochastic Integrals where X is an (IHI, Qn) semimartingale for each n. Therefore by Theorem 3 we conclude X is an (IHI, P) semimartingale. 0 Corollary. Let A be a finite collection of events in F, and let IHI = (1t t h?:.o be the filtration generated by Ft and A. Then every (IF, P) semimartingale is an (IHI, P) semimartingale also. Proof. Since A is finite, one can always find a (finite) partition II of n such that 1tt = F t V II. The corollary then follows by Theorem 5. 0 Note that if B = (Bt)t?:.o is a Brownian motion for a filtration (Ft)t?:.o, by Theorem 5 we are able to add, in a certain manner, an infinite number of "future" events to the filtration and B will no longer be a martingale, but it will stay a semimartingale. This has interesting implications in finance theory (the theory of continuous trading). See for example Duffie-Huang [60]. The corollary of the next theorem states that being a semimartingale is a "local" property; that is, a local semimartingale is a semimartingale. We get a stronger result by stopping at Tn - rather than at Tn in the next theorem. A process X is stopped at T- if Xi- = Xtl{oS:;t O. Define R n = Tnl{Tns:;t} + 00l{Tn>t}. Then But P(Rn < 00) = P(Tn ::;; t), and since Tn increases to 00 a.s., P(Tn ::;; t) --+ 0 as n --+ 00. Thus if Hk tends to 0 in Su, given E > 0, we choose n so that P(Rn < 00) < E/2, and then choose k so large that P{II(xn)t(Hk)1 ::::: c} < E/2. Thus, for k large enough, P{IIxt(Hk)1 ::::: c} < E. 0 Corollary. Let X be a process. If there exists a sequence (Tn) of stopping times increasing to 00 a.s., such that XTn (or X Tn l{Tn>O}) is a semimartin- gale, each n, then X is also a semimartingale. 3 Elementary Examples of Semimartingales The elementary properties of semimartingales established in Sect. 2 will allow us to see that many common processes are semimartingales. For example, the Poisson process, Brownian motion, and more generally all Levy processes are semimartingales. 3 Elementary Examples of Semimartingales 55 Theorem 7. Each adapted process with cddldg paths of finite variation on compacts (of finite total variation) is a semimartingale (a total semimartin- gale). Proof. It suffices to observe that IIx(H)1 'S IIHllu I:' IdXsl, where Iooo IdXsl denotes the Lebesgue-Stieltjes total variation and IIHllu = sup(t,w) IH(t,w)l. o Theorem 8. Each L 2 martingale with cddldg paths is a semimartingale. Proof. Let X be an L2 martingale with X o = 0, and let H E S. Using Doob's Optional Sampling Theorem and the L2 orthogonality of the increments of L 2 martingales, it suffices to observe that n n E{(Ix(H))2} = E{(I: Hi (XTi+l - X TJ)2} = E{I: Hl(XTi+1 - X TJ 2} i=O i=O n n 'S IIHII~E{I:(XTi+l - XTY} = IIHII~E{I:(Xf,i+l - Xf,J} i=O i=O o Corollary 1. Each cadlag, locally square integrable local martingale is a semimartingale. Proof. Apply Theorem 8 together with the corollary to Theorem 6. 0 Corollary 2. A local martingale with continuous paths is a semimartingale. Proof. Apply Corollary 1 together with Theorem 51 in Chap. 1. o Corollary 3. The Wiener process (that is, Brownian motion) is a semimartin- gale. Proof. The Wiener process B t is a martingale with continuous paths if B o is integrable. It is always a continuous local martingale. 0 Definition. We will sayan adapted process X with cadlag paths is decom- posable if it can be decomposed Xt = Xo + Mt + At, where Mo = Ao = 0, M is a locally square integrable martingale, and A is cadlag, adapted, with paths of finite variation on compacts. Theorem 9. A decomposable process is a semimartingale. Proof. Let X t = Xo + Mt + At be a decomposition of X. Then M is a semi- martingale by Corollary 1 of Theorem 8, and A is a semimartingale by The- orem 7. Since semimartingales form a vector space (Theorem 1) we have the result. 0 Corollary. A Levy process is a semimartingale. 56 II Semimartingales and Stochastic Integrals Proof By Theorem 40 of Chap. I we know that a Levy process is decompos- able. Theorem 9 then gives the result. D Since Levy processes are prototypic strong Markov processes, one may well wonder if alllRn-valued strong Markov processes are semimartingales. Simple examples, such as X t = B;/3, where B is standard Brownian motion, show this is not the case (while this example is simple, the proof that X is not a semimartingale is not elementaryl ). However if one is willing to "regularize" the Markov process by a transformation of the space (in the case of this example using the "scale function" S(x) = x 3 ), "most reasonable" strong Markov processes are semimartingales. Indeed, Dynkin's formula, which states that if f is in the domain of the infinitesimal generator G of the strong Markov process Z, then the process is well-defined and is a local martingale, hints strongly that if the domain of G is rich enough, the process Z is a semimartingale. In this regard see Sect. 7 of Qinlar, Jacod, Protter, and Sharpe [34]. 4 Stochastic Integrals In Sect. 1 we defined semimartingales as adapted, cadlag processes that acted as "good integrators" on the simple predictable processes. We now wish to enlarge the space of processes we can consider as integrands. In Chap. IV we will consider a large class of processes, namely those that are "predictably measurable" and have appropriate finiteness properties. Here, however, by keeping our space of integrands small-yet large enough to be interesting- we can keep the theory free of technical problems, as well as intuitive. A particularly nice class of processes for our purposes is the class of adapted processes with left continuous paths that have right limits (the French acronym would be caglad). Definition. We let j[J) denote the space of adapted processes with cadlag paths, II... denote the Space of adapted processes with caglad paths (left con- tinuous with right limits) and blI... denote processes in II... with bounded paths. We have previously considered Su, the space of simple, predictable pro- cesses endowed with the topology of uniform convergence; and LO, the space of finite-valued random variables topologized by convergence in probability. We need to consider a third type of convergence. I See Theorem 71 of Chap. IV which proves a similar assertion for X t = IBtl", 0< a < 1/2. 4 Stochastic Integrals 57 Definition. A sequence of processes (Hn)n?l converges to a process H uni- formly on compacts in probability (abbreviated ucp) if, for each t > 0, sUPo~s~t IH~ - Hsi converges to 0 in probability. We write H; = sUPo n}. Then R n is a stopping time and yn = yRn I{Rn>o} are in bL and converge to Y in ucp. Thus bL is dense in L. Without loss we now assume Y E bL. Define Z by Zt = limu.....t Yu' Then u>t Z E lDl. For c > 0, define Tg =0 T~+l = inf{t: t > T~ and IZt - ZT~I > c}. Since Z is dtdlag, the T~ are stopping times increasing to 00 a.s. as n increases. Let ze = "'n ZT O. Then ze are bounded and con-o n n' n+l verge uniformly to Z as c tends to O. Let Ue = }Ol{o} + ~n ZT~I(T~,T~+lJ' Then U e E bL and the preceding implies U e converges uniformly on compacts to Yol{o} + Z_ = Y. Finally, define n yn,e = }QI{o} + '" ZT 58 II Semimartingales and Stochastic Integrals Definition. For H E S and X a cadlag process, define the (linear) mapping JX : S ---+ J!)) by n Jx(H) = HoXo + 2: Hi (XTi+l - XTi) i=l for H in S with the representation n H = Hol{o} + LHil(Ti,T,+lj, i=l Hi E FTi and 0 = To s:; T1 s:; ... s:; Tn+1 < 00 stopping times. Definition. For H E S and X an adapted cadlag process, we call Jx(H) the stochastic integral of H with respect to X. We use interchangeably three notations for the stochastic integral: ;x(H) = JHsdXs = H· X. Observe that Jx(H)t = Jxt(H). Indeed, Ix plays the role of a definite inte- gral. For H E S, Jx(H) = 1000 HsdXs. Theorem 11. Let X be a semimartingale. Then the mapping Jx : Sucp ---+ [])ucp is continuous. Proof Since we are only dealing with convergence on compact sets, without loss of generality we take X to be a total semimartingale. First suppose Hk in S tends to 0 uniformly and is uniformly bounded. We will show Jx(Hk) tends to 0 ucp. Let J > 0 be given and define stopping times T k by T k = inf{t : I(H k . X)tl ~ J}. Then H k l[o,Tkj E S and tends to 0 uniformly as k tends to 00. Thus for every t, P{(Hk . X); > J} s:; P{IHk . XTk/\tl ~ J} = P{I(Hk l[o,Tkj . Xhl ~ J} = P{IIx(Hk l[o,Tk/\tj) I ~ J} which tends to 0 by the definition of total semimartingale. We have just shown that Jx : Su ---+ J!))ucp is continuous. We now use this to show Jx : Sucp ---+ J!))ucp is continuous. Suppose H k goes to 0 ucp. Let J > 0, c > 0, t > O. We have seen that there exists 'I} such that IIHllu s:; 'I} implies P(Jx(H); > J) < i. Let Rk = inf{s : IH:I > 'I}}, and set jjk = H kl[o,Rkjl{Rk>o}, Then jjk E Sand IIjjkll u s:; 'I} by left continuity. Since Rk ~ t implies (jjk . X); = (Hk . X);, we have 4 Stochastic Integrals 59 P((H k . X); > J) ::::; p((iik . X); > J) + P(Rk < t) ::::; ~ + P((Hk ); > 1)) 1)) = O. D We have seen that when X is a semimartingale, the integration operator Jx is continuous on Sucp, and also that Sucp is dense in L ucp . Hence we are able to extend the linear integration operator JX from S to L by continuity, since [])ucp is a complete metric space. Definition. Let X be a semimartingale. The continuous linear mapping Jx : L ucp ---> [])ucp obtained as the extension of Jx : S ---> []) is called the stochastic integral. The preceding definition is rich enough for us to give an immediate example of a surprising stochastic integral. First recall that if a process (Adt~o has continuous paths of finite variation with Ao = 0, then the Riemann-Stieltjes integral of J; AsdAs yields the formula (see Theorem 54 of Chap. I) i t 1AsdAs = -A;.o 2 Let us now consider a standard Brownian motion B = (Bdt>o with B o = O. The process B does not have paths of finite variation on compacts, but it is a semimartingale. Let (7rn) be a refining sequence of partitions of [0, 00) with limn -+oo mesh(7rn ) = O. Let Bf = LtkE7I"n B tk l(tk,tk+l]' Then Bn E L for each n. Moreover, Bn converges to B in UCfJ. Fix t :::: 0 and assume that t is a partition point of each 7rn . Then JB(Bn)t = L B tk (Btk+l _Btk) tkE7I"n tk 60 II Semimartingales and Stochastic Integrals i t 1 1BsdBs = -B; - -t,0 22 a formula distinctly different from the Riemann-Stieltjes formula (*). The change of variables formula (Theorem 32) presented in Sect. 7, will give a deeper understanding of the difference in formulas (*) and (**). 5 Properties of Stochastic Integrals Throughout this paragraph X will denote a semimartingale and H will denote an element of L. Recall that the stochastic integral defined in Sect. 4 will be denoted by the three notations Jx(H) = H· X = J HsdXs. Evaluating these processes at t, we have H . X t = r HsdXs = r HsdXs.Jo J[O,tJ To exclude 0 in the integral we write rt HsdXs = r HsdXs. Jo+ JeO,tJ The integral Jooo HsdXs is defined to be limt-+oo J~ HsdXs when the limit exists. Note that J~ HsdXs = HoXo + J~+ HsdXs. For a process Y E lDl, we recall that ~yt = yt - yt_, the jump at t. Also since Yo- = 0 we have ~Yo = Yo· Recall further that for a process Z and stopping time T, we let ZT denote the stopped process (Zr) = ZtIlT. We will establish in this section several elementary properties of the stochastic integral. These properties will help us to understand the integral and to examine examples at the end of the section. Recall that two processes Y and Z are indistinguishable if P{w: t f---' Xt(w) and t f---' yt(w) are the same functions} = 1. Theorem 12. Let T be a stopping time. Then (H. X)T = Hl[o,Tj . X = H. (XT ). Theorem 13. The jump process ~(H.X)s is indistinguishable from Hs(~Xs). Proofs. Both properties are clear when H E S, and they follow when HElL by passing to the limit. Indeed, we take convergence in ucp for each t, then a.s. convergence by taking a subsequence, and then we choose the rationals (which of course is countable and dense, so the union of null sets is still a null set), and then use the path regularity to get indistinguishability. D Let Q denote another probability law, and let HQ.X denote the stochastic integral of H with respect to X computed under the law Q. Theorem 14. Let Q « P. Then H Q . X is Q indistinguishable from H p . x. 5 Properties of Stochastic Integrals 61 Proof Note that by Theorem 2, X is a Q semimartingale. The theorem is clear if H E S, and it follows for H E L by passage to the limit in the ucp topology, since convergence in P-probability implies convergence in Q-probability. D Theorem 15. Let Pk be a sequence of probabilities such that X is a Pk semimartingale for each k. Let R = L:r=l AkPk where Ak ~ 0, each k, and L:~1 Ak = 1. Then HR· X = Hpk . X, Pk-a.s., for all k such that Ak > o. Proof If Ak > 0 then Pk ~ R, and the result follows by Theorem 14. Note that by Theorem 3 we know that X is an R semimartingale. D Corollary. Let P and Q be any probabilities and suppose X is a semimartin- gale relative to both P and Q. Then there exists a process H . X which is a version of both Hp . X and HQ . x. Proof Let R = (P+Q)/2. Then HR·X is such a process by Theorem 15. D Theorem 16. Let G = Wdt>o be another filtration such that H is in both L(G) and L(lF'), and such that X is also a G semimartingale. Then HG· X = HF·X. Proof L(G) denotes left continuous processes adapted to the filtration G. As in the proof of Theorem 10, we can construct a sequence of processes Hn converging to H where the construction of the H n depends only on H. Thus Hn E S(G) n S(lF') and converges to H in ucp. Since the result is clear for S, the full result follows by passing to the limit. D Remark. While Theorem 16 is a simple result in this context, it is far from simple if the integrands are predictably measurable processes, rather than processes in L. See the comment following Theorem 33 of Chap. IV. The next two theorems are especially interesting because they show-at least for integrands in lL-that the stochastic integral agrees with the path- by-path Lebesgue-Stieltjes integral, whenever it is possible to do so. Theorem 17. If the semimartingale X has paths of finite variation on com- pacts, then H· X is indistinguishable from the Lebesgue-Stieltjes integral, com- puted path-by-path. Proof The result is evident for H E S. Let H n E S converge to H in ucp. Then there exists a subsequence nk such that limnk .....oo(Hnk - H); = 0 a.s., and the result follows by interchanging limits, justified by the uniform a.s. convergence. D Theorem 18. Let X, X be two semimartingales, and let H, H E L. Let A = {w : H.(w) = H.(w) and x.(w) = X(w)}, and let B = {w : t f---' Xt(w) is of finite variation on compacts}. Then H . X = H . X on A, and H . X is equal to a path-by-path Lebesgue-Stieltjes integral on B. 62 II Semimartingales and Stochastic Integrals Proof. Without loss of generality we assume P(A) > O. Define a new prob- ability law Q by Q(A) = P(AIA). Then under Q we have that H and H as well as X and X are indistinguishable. Thus H Q . X = H Q . X, and hence H . X = H . X P-a.s. on A by Theorem 14, since Q ~ P. As for the second assertion, if B = n the result is merely Theorem 17. Define R by R(A) = P(AIB), assuming without loss that P(B) > O. Then R ~ P and B = n, R-a.s. Hence HR' X equals the Lebesgue-Stieljes integral R-a.s. by Theorem 17, and the result follows by Theorem 14. 0 The preceding theorem and following corollary are known as the local behavior of the integral. Corollary. With the notation of Theorem 18, let S, T be two stopping times with S < T. Define C = {w : Ht(w) = Ht(w); Xt(w) = Xt(w); S(w) < t s:; T(w)} D = {w : t f---' Xt(w) is of finite variation on S(w) < t < T(w)}. Then H . X T - H . XS = H . X T - H . X S on C and H . X T - H . XS equals a path-by-path Lebesgue-Stieltjes integral on D. Proof. Let yt = X t - XtIlS. Then H . Y = H . X - H . XS, and Y does not change the set [0, S], which is evident, or which-alternatively-can be viewed as an easy consequence of Theorem 18. One now applies Theorem 18 to yT to obtain the result. 0 Theorem 19 (Associativity). The stochastic integral process Y = H· X is itself a semimariingale, and for GEL we have G· Y = G· (H· X) = (GH) . X. Proof. Suppose we know Y = H· X is a semimartingale. Then G· Y = Jy (G). If G, H are in S, then it is clear that Jy(G) = Jx(GH). The associativity then extends to L by continuity. It remains to show that Y = H· X is a semimartingale. Let (Hn) be in S converging in ucp to H. Then Hn . X converges to H· X in ucp. Thus there exists a subsequence (nk) such that Hnk . X converges a.s. to H . X. Let G E S and let ynk = Hnk .X, Y = H· X. The ynk are semimartingales converging pointwise to the process Y. For G E S, Jy(G) is defined for any process Y; so we have Jy(G)=G.Y= lim G·ynk= lim G·(Hnk.X) nk~OO nk~OO = lim (GHnk). X nk~OO which equals limnk -+oo Jx(GHnk) = Jx(GH), since X is a semimartingale. Therefore Jy(G) = Jx(GH) for G E S. 5 Properties of Stochastic Integrals 63 Let en converge to e in SUo Then enH converges to eH in Lucp , and since X is a semimartingale, limn -+oo Jy (en) = limn -+oo Jx (en H) = Jx (eH) = Jy (e). This implies yt is a total semimartingale, and so Y = H . X is a semimartingale. D Theorem 19 shows that the property of being a semimartingale is preserved by stochastic integration . Also by Theorem 17 if the semimartingale X is an FV process, then the stochastic integral agrees with the Lebesgue-Stieltjes integral, and by the theory of Lebesgue-Stieltjes integration we are able to conclude the stochastic integral is an FV process also. That is, the property of being an FV process is preserved by stochastic integration for integrands in L.2 One may well ask if other properties are preserved by stochastic integra- tion; in particular, are the stochastic integrals of martingales and local mar- tingales still martingales and local martingales? Local martingales are indeed preserved by stochastic integration, but we are not yet able easily to prove it. Instead we show that locally square integrable local martingales are preserved by stochastic integration for integrands in L. Theorem 20. Let X be a locally square integrable local martingale, and let H E L. Then the stochastic integral H . X is also a locally square integrable local martingale. Proof We have seen that a locally square integrable local martingale is a semimartingale (Corollary 1 of Theorem 8), so we can formulate H . X. With- out loss of generality, assume X o = 0. Also, if Tk increases to 00 a.s. and (H· X)Tk is a locally square integrable local martingale for each k, it is simple to check that H . X itself is one. Thus without loss we assume X is a square integrable martingale. By stopping H, we may further assume H is bounded, by fl.. Let Hn E S be such that Hn converges to H in ucp. We can then modify Hn, call it jjn, such that jjn is bounded by fl., jjn E S, and jjn converges uniformly to H in probability on [0, tJ. Since jjn E bS, one can check that jjn . X is a martingale. Moreover kn E{(jjn. X);} = E(L(jj['(X;i+l - Xr))2} i=l kn ::; fl.2E{2:(XLl - X;,J} i=l ::; fl.2E{X~}, and hence (jjn . X)t are uniformly bounded in L 2 and thus uniformly inte- grable. Passing to the limit then shows both that H . X is a martingale and that it is square integrable. D 2 See Exercise 43 in Chap. IV which shows this is not true in general. 64 II Semimartingales and Stochastic Integrals In Theorem 29 of Chap. III we show the more general result that if M is a local martingale and H E JL, then H . M is again a local martingale. A classical result from the theory of Lebesgue measure and integration (on lR) is that a bounded, measurable function f mapping an interval [a, b] to lR is Riemann integrable if and only if the set of discontinuities of f has Lebesgue measure zero (e.g., Kingman and Taylor [127, page 129]). Therefore we cannot hope to express the stochastic integral as a limit of sums unless the integrands have reasonably smooth sample paths. The spaces lDl and JL consist of processes which jump at most countably often. As we will see in Theorem 21, this is smooth enough. Definition. Let u denote a finite sequence of finite stopping times: The sequence u is called a random partition. A sequence of random parti- tions un, Un: Tf/ s:; T'{' s:; . . . s:; Tkn is said to tend to the identity if (i) limn sUPk Tk = 00 a.s.; and (ii) Ilunll = sUPk ITk+l - Tkl converges to °a.s. Let Y be a process and let u be a random partition. We define the process Y sampled at u to be yO" == Yol{o} + LYTkl(Tk,Tk+l]' k It is easy to check that f YsO" dXs = YoXo + 2.:= YTi (XTi+l - XTi), • for any semimartingale X, any process Y in S, lDl, or JL. Theorem 21. Let X be a semimartingale, and let Y be a process in lDl or in JL. Let (un) be a sequence of random partitions tending to the identity. Then the processes J~+ YsO"ndXs = 2:i YT:: (X{+l - x;t) tend to the stochastic integral (L)· X in ucp. Proof (The notation y_ means the process whose value at s is given by (Y_)s = limu.....s,u 5 Properties of Stochastic Integrals 65 J(L - ylTn)sdXs = f (L - yk)sdXs + f (yk - (Yt)lTn)sdXs +J((y1n lTn - YlTn )sdXs where y-t denotes the cadlag version of yk. The first term on the right side equals Jx(Y- _yk), and since Jx is continuous in Lucp and since y _ _ yk --t 0, we have J(Y- - yk)sdXs --t 0 (ucp). The same reasoning applies to the third term, for fixed n, as k --t 00. Indeed, the convergence to 0 of ((y-t)lTn - YlTn) as k --t 00 is uniform in n. It remains to consider the middle term on the right side above. Since the yk are simple predictable we can write the stochastic integrals in closed form, and since X is right continuous the integrals (for fixed (k, w)) J(yk - (Y-t)lTn)sdXs tend to 0 as n --t 00. Thus one merely chooses k so large that the first and third terms are small, and then for fixed k, the middle term can be made small for large enough n. 0 We consider here another example. We have already seen at the end of Sect. 4 that if B is a standard Wiener process, then i t 1 1BsdBs = -B; - -t,0 22 showing that the semimartingale calculus does not formally generalize the Riemann-Stieltjes calculus. We have seen as well that the stochastic integral agrees with the Lebesgue- Stieltjes integral when possible (Theorems 17 and 18) and that the stochastic integral also preserves the martingale property (Theorem 20; at least for lo- cally square integrable local martingales). The follOWing example shows that the restriction of integrands to L is not as innocent as it may seem if we want to have both of these properties. Example. Let M t = Nt - >.t, a compensated Poisson process (and hence a martingale with M t E LP for all t ~ 0 and all p ~ 1). Let (Ti k::1 be the jump times of M. Let H t = I[O,T,)(t). Then H E ID>. The Lebesgue-Stieltjes integral is It HsdMs = It HsdNs - >.It Hsds 00 t = L HTi l{t::;':Ti} - >.1 Hsds i=1 0 = ->.(t 1\ TI). 66 II Semimartingales and Stochastic Integrals This process is not a martingale. We conclude that the space of integrands cannot be expanded even to lDl, in general, and preserve the structure of the theory already established.3 6 The Quadratic Variation of a Semimartingale The quadratic variation process of a semimartingale, also known as the bracket process, is a simple object that nevertheless plays a fundamental role. Definition. Let X, Y be semimartingales. The quadratic variation pro- cess of X, denoted [X,X] = ([X,X]t}t2:0, is defined by [X, X] = X 2 - 2JX_dX (recall that Xo- = 0). The quadratic covariation of X, Y, also called the bracket process of X, Y, is defined by [X,Y] = XY - JX_dY - JY-dX. It is clear that the operation (X, Y) --t [X, Y] is bilinear and symmetric. We therefore have a polarization identity 1[X,Y] = 2([X + Y,X + Y] - [X, X] - [Y,YJ). The next theorem gives some elementary properties of [X,X]. (X is assumed to be a given semimartingale throughout this section). Theorem 22. The quadratic variation process of X is a cddldg, increasing, adapted process. Moreover it satisfies the following. (i) [X,X]o = X5 and ~[X,X] = (~X)2. (ii) If (Tn is a sequence of random partitions tending to the identity, then with convergence in ucp, where (Tn is the sequence 0 = T?J' s:: Tf s:: ... s:: Tr s:: ... s:: Tkn and where Tr are stopping times. (iii) If T is any stopping time, then [XT, X] = [X, XT] = [XT , XT] = [X,X]T. 3 Ruth Williams has commented to us that this example would be more convincing if M were itself a semimartingale, H bounded and in JI)), and J; HsdMs were not a semimartingale. Such a construction is carried out in [1]. 6 The Quadratic Variation of a Semimartingale 67 Proof X is cadlag, adapted, and so also is JX_dX by its definition; thus [X, Xl is cadlag, adapted as well. Recall the property of the stochastic integral: ~(X_· X) = X_~X. Then (~X); = (Xs - X s_)2 = X; - 2XsX s- + X;_ = X; - X;_ +2Xs-(Xs- - X s ) = ~(X2)s - 2Xs-(~Xs), from which part (i) follows. ~ For part (ii), by replacing X with X = X - X o, we may assume X o = O. Let Rn = SUPi Tt. Then Rn < 00 a.s. and limn Rn = 00 a.s., and thus by telescoping series converges ucp to X 2 . Moreover, the series Li XTt(XTinj-l _xTt) converges in ucp to JX_dX by Theorem 21, since X is cadlag. Since b2 - a2 - 2a(b - a) = (b-a)2, and since XTn(XTt+l _xTt) = XTt(XT['j-l-XTt), we can combine the two series convergences above to obtain the result. Finally, note that if s < t, then the approximating sums in part (ii) include more terms (all non- negative), so it is clear that [X, X] is non-decreasing. (Note that, a priori, one only has [X, X]s ::; [X, X]t a.s., with the null set depending on sand t; it is the property that [X, X] has cadlag paths that allows one to eliminate the dependence of the null set on sand t.) Part (iii) is a simple consequence of ~rt(~. 0 An immediate consequence of Theorem 22 is the observation that if B is a Brownian motion, then [B, Blt = t, since in Theorem 28 of Chap. I we showed the a.s. convergence of sums of the form in part (ii) of Theorem 22 when the partitions are refining. Another consequence of Theorem 22 is that if X is a semimartingale with continuous paths of finite variation, then [X, X] is the constant process equal to XJ. To see this one need only observe that 2)XTi+l - X Tt)2 ::; sup IXTi+l - XTinIL IXTi+l - xTt I , ::; sup IXTi+l - xTt IV, i where V is the total variation. Therefore the sums tend to 0 as II(Tnll ----t O. Theorem 22 has several more consequences which we state as corollaries. Corollary 1. The bracket process [X, Yl of two semimartingales has paths of finite variation on compacts, and it is also a semimartingale. 68 II Semimartingales and Stochastic Integrals Proof. By the polarization identity [X, Yj is the difference of two increas- ing processes, hence its paths are of finite variation. Moreover, the paths are clearly cadlag, and the process is adapted. Hence by Theorem 7 it is a semi- martingale. 0 Corollary 2 (Integration by Parts). Let X, Y be two semimartingales. Then X Y is a semimartingale and XY = JX_dY +JLdX + [X, Yj. Proof. The formula follows trivially from the definition of [X, Y]. That XY is a semimartingale follows from the formula, Theorem 19, and Corollary 1 ~~. 0 In the integration by parts formula above, we have (X_)o = (Y_)o = O. Hence evaluating at 0 yields XoYo = (X_)oYo + (L)oXo + [X, Yjo. Since [X, Y]o = ~Xo~Yo = XoYo, the formula is valid. Without the conven- tion that (X_)o = 0, we could have written the formula Corollary 3. All semimartingales on a given filtered probability space form an algebra. Proof. Since semimartingales form a vector space, Corollary 2 shows they form an algebra. 0 A theorem analogous to Theorem 22 holds for [X, Y] as well as [X,X]. It can be proved analogously to Theorem 22, or more simply by polarization. We omit the proof. Theorem 23. Let X and Y be two semimartingales. Then the bracket process [X, Y] satisfies the following. (i) [X, Y]o = XoYo and ~[X, Y] = ~X~Y. (ii) If Un is a sequence of random partitions tending to the identity, then [X, Yj = XoYo + lim "'(XT;'+l - X Tt)(yT;'+l _ yTt), n--+oo L...J i where convergence is in ucp, and where Un is the sequence 0 = To :s: Tf :s: ... :s: ~n :s: ... :s: Tkn , with Tt stopping times. (iii) 1fT is any stopping time, then [XT, Y] = [X, y T ] = [XT, yTj = [X, YjT. 6 The Quadratic Variation of a Semimartingale 69 We next record a real analysis theorem from the Lebesgue-Stieltjes theory of integration. It can be proved via the Monotone Class Theorem. Theorem 24. Let 0:, (3, "'( be functions mapping [0,00) to lR with 0:(0) = (3(0) = "'(0) = O. Suppose 0:, (3, "'( are all right continuous, 0: is of finite variation, and (3 and "'( are each increasing. Suppose further that for all s, t with s :S t, we have Then for any measurable functions f, g we have In particular, the measure do: is absolutely continuous with respect to both d(3 andd"'(. Note that Ido:l denotes the total variation measure corresponding to the measure do:, the Lebesgue-Stieltjes signed measure induced by 0:. We use this theorem to prove an important inequality concerning the quadratic variation and bracket processes. Theorem 25 (Kunita-Watanabe Inequality). Let X and Y be two semi- martingales, and let Hand K be two measurable processes. Then one has a.s. Proof By Theorem 24 we only need to show that there exists a null set N, such that for w rJ. N, and (s, t) with s :S t, we have Let N be the null set such that if w rJ. N, then 0 :S f: d[X + rY, X + rY]u, for every r, s, t; s :S t, with r, s, t all rational numbers. Then 0:S [X + rY,X + rY]t - [X + rY,X + rY]s = r 2 ([y, Y]t - [Y, Y]s) + 2r([X, Y]t - [X, Y]s) + ([X, X]t - [X, X]s). The right side being positive for all rational r, it must be positive for all real r by continuity. Thus the discriminant of this quadratic equation in r must be non-negative, which gives us exactly the inequality (*). Since we have, then, the inequality for all rational (s, t), it must hold for all real (s, t), by the right continuity of the paths of the processes. D 70 II Semimartingales and Stochastic Integrals Corollary. Let X and Y be two semimartingales, and let Hand K be two measurable processes. Then if .! + .! = 1.p q Proof Apply Holder's inequality to the Kunita-Watanabe inequality of The- orem 25. 0 Since Theorem 25 and its corollary are path-by-path Lebesgue-Stieltjes results, we do not have to assume that the integrand processes Hand K be adapted. Since the process [X, Xl is non-decreasing with right continuous paths, and since ~[X,Xlt = (~Xt)2 for all t 2: 0 (with the convention that X o- = 0), we can decompose [X, X] path-by-path into its continuous part and its pure jump part. Definition. For a semimartingale X, the process [X, X] C denotes the path- by-path continuous part of [X,X]. We can then write [X, Xl t = [X, X]f + XJ + L (~Xs)2 o 6 The Quadratic Variation of a Semimartingale 71 Definition. A semimartingale X will be called quadratic pure jump if [X,X]C = O. If X is quadratic pure jump, then [X, X]t = XJ + L:o 72 II Semimartingales and Stochastic Integrals Proof. Note that a continuous local martingale is a semimartingale (Corol- lary 2 of Theorem 8). We have X2 - [X,Xl = 2J X_dX, and by the mar- tingale preservation property (Theorem 20) we have that 2 JX_dX is a local martingale. Moreover ~2 JX_dX = 2(X_ )(~X), and since X is continu- ous, ~X = 0, and thus 2 JX_dX is a continuous local martingale, hence locally square integrable. Thus X 2 - [X, X] is a locally square integrable local martingale. By stopping, we can suppose X is a square integrable martingale. As- sume further X o = O. Next assume that [X, Xl actually were constant. Then [X,X]t = [X,Xlo = X5 = 0, for all t. Since X 2 - [X,Xl is a local mar- tingale, we conclude X 2 is a non-negative local martingale, with XJ = O. Thus xl = 0, all t. This is a contradiction. If X o is not identically 0, we set Xt = X t - X o and the result follows. 0 The following corollary is of fundamental importance in the theory of martingales. Corollary 1. Let X be a continuous local martingale, and 8 :s; T :s; 00 be stopping times. If X has paths of finite variation on the stochastic interval (8, T), then X is constant on [8, T]. Moreover if [X, X] is constant on [8, T] n [0,00), then X is constant there too. Proof. M = XT - X S is also a continuous local martingale, and M has finite variation on compacts. Moreover [M,M] = [XT - XS,XT - XS] = [X,X1 T - [X,X1 S, and therefore [M,M] is constant everywhere, hence by Theorem 27, M must be constant everywhere; thus X is constant on [8, T]. 0 Observe that if t > 0 is arbitrary and we take 8 = 0 and T = t in Corol- lary 1, then we can conclude that a continuous local martingale with paths of finite variations on compacts is a.s. constant. While non-trivial continu- ous local martingales must therefore always have paths of infinite variation on compacts, they are not the only such local martingales. For example, the Levy process martingale Zd of Theorem 41 of Chap. I will have paths of infi- nite variation if the Levy measure 1/ has infinite mass in a neighborhood of the origin. Another example is Azema's martingale, which is presented in detail in Sect. 8 of Chap. IV. Corollary 2. Let X and Y be two locally square integrable local martin- gales. Then [X, Y] is the unique adapted cadlag process A with paths of finite variation on compacts satisfying the two properties: (i) XY - A is a local martingale; and (ii) ~A = ~X~Y, Ao = XoYo. Proof. Integration by parts yields XY = JX_dY +JLdX + [X, Y], 6 The Quadratic Variation of a Semimartingale 73 but the martingale preservation property tells us that both stochastic integrals are local martingales. Thus XY - [X, Y] is a local martingale. Property (ii) is simply an application of Theorem 23. Thus it remains to show uniqueness. Suppose A, B both satisfy properties (i) and (ii). Then A - B = (XY - B) - (XY - A), the difference of two local martingales which is again a local martingale. Moreover, ~(A - B) = ~A - ~B = ~X~Y - ~X~Y = o. Thus A - B is a continuous local martingale, Ao - Bo = 0, and it has paths of finite variation on compacts. Corollary 1 yields At - Bt - Ao+ Bo = 0 and we have uniqueness. 0 Corollary 2 can be useful in determining the process [X, Y]. For example if X and Yare locally square integrable martingales without common jumps such that XY is a martingale, then [X, Y] = XoYo. One can also easily verify (as a consequence of Theorem 23 and Corollaries 1 and 2 above, for example) that if X is a continuous square integrable martingale and Y is a square integrable martingale with paths of finite variation on compacts, then [X, Y] = XoYo, and hence XY is a martingale. (An example would be X a Brownian motion and Y a compensated Poisson process.) Corollary 2 is true as well for X, Y local martingales, however we need Theorem 29 of Chap. III to prove the general result. In Chap. III we show that any local martingale is a semimartingale (see the corollary of Theorem 26 of Chap. III), and therefore if M is a local martingale its quadratic variation [M, Mlt always exists and is finite a.s. for every t 2: o. We use this fact in the next corollary. Corollary 3. Let M be a local martingale. Then M is a martingale with E{Mn < 00, all t 2: 0, if and only if E{ [M, M]t} < 00, all t 2: O. If E{[M,M]t} < 00, then E{Mn = E{[M,M]d. Proof First assume that M is a martingale with E{Mn < 00 for all t 2: o. Then M is clearly a locally square integrable martingale. Let Nt = Ml - [M, M]t = ~ J~ Ms_dMs, which is a locally square integrable local martingale by Theorem 20. Let (Tn)n>l be stopping times increasing to 00 a.s. such that Nt n is a square integr~hlemartingale. Then E{Nr} = E{No} = 0, all t 2: O. Therefore E{Ml,'\Tn} = E{[M,M]t!\Tn}. Doob's maximal quadratic inequality gives E{(Mn 2 } < 4E{Mn < 00. Therefore by the Dominated Convergence Theorem E{Mt2 } = lim E{Mt2!\Tn}n--->oo = lim E{[M, M]t!\Tn} n--->oo = E{[M, M]t}, 74 II Semimartingales and Stochastic Integrals where the last result is by the Monotone Convergence Theorem. In particular we have that E{[M, M]t} < 00. For the converse, we now assume E{[M, M]t} < 00, all t 2: O. Define stopping times by Tn = inf{t > 0: IMtl > n} An. Then Tn increase to 00 a.s. Furthermore (MTn )* ::; n + I~MTnl ::; n + [M, M];/2, which is in £2. By Theorem 51 of Chap. I, M Tn is a uniformly integrable martingale for each n. Also we have that E{(Mr)2} ::; E{((MTn )*)2} < 00, for all t 2: O. Therefore M Tn satisfies the hypotheses of the first half of this theorem, and E{(Mr)2} = E{[MTn ,MTn]t}. Using Doob's inequality we have 2 Tn 2 Tn TnE{(Mtf\Tn) } ::; 4E{(Mt )} = 4E{[M ,M Jt} = 4E{[M, M]Tnf\t} ::; 4E{[M, M]t}. The Monotone Convergence Theorem next gives E{(Mt)2} = lim E{(Mtf\Tn)2} n->oo ::; 4E{[M, M]t} < 00. Therefore, again by Theorem 51 of Chap. I, we conclude that M is a martin- gale. The preceding gives E{Ml} < 00. 0 For emphasis we state as another corollary a special case of Corollary 3. Corollary 4. If M is a local martingale and E{[M,M]oo} < 00, then M is a square integrable martingale (that is SUPt E {M?} = E {M~} < 00). Moreover E{Ml} = E{[M, M]t} for all t, 0 ::; t ::; 00. Example. Before continuing we consider again an example of a local martin- gale that exhibits many of the surprising pathologies of local martingales. Let B be a standard Brownian motion in]R3 with Bo = (1,1,1). Let M t = IIBtll- 1, where Ilxll is standard Euclidean norm in ]R3. (We previously considered this example in Sect. 6 of Chap. I.) As noted in Chap. I, the process M is a contin- uous local martingale; hence it is a locally square integrable local martingale. Moreover E{Ml} < 00 for all t. However instead of t f---t E{Ml} being an increasing function as it would if M were a martingale, limt->oo E{Ml} = O. Moreover E{[M, M]t} 2: E{[M, M]o} = 1 since [M, M]t is increasing. There- fore we cannot have E{Ml} = E{[M,M]t} for all t. Indeed, by Corollary 3 and the preceding we see that we must have E{[M, M]t} = 00 for all t > O. In conclusion, M = IIBII- 1 is a continuous local martingale with E{Ml} < 00 for all t which is both not a true martingale and for which E{Ml} < 00 while E{[M,M]t} = 00 for all t > O. (Also refer to Exercise 20 at the end of this chapter.) 6 The Quadratic Variation of a Semimartingale 75 Corollary 5. Let X be a continuous local martingale. Then X and [X, X] have the same intervals of constancy a.s. Proof Let r be a positive rational, and define Then M = XTr - xr is a local martingale which is constant. Hence [M, M] = [X, X]Tr - [X, xt is also constant. Since this is true for any rational r a.s., any interval of constancy of X is also one of [X, X]. Since X is continuous, by stopping we can assume without loss of generality that X is a bounded martingale (and hence square integrable). For every positive, rational r we define Sr = inf{ t 2: r : [X, X]t > [X, X]r}. Then E{(XSr - X r )2} = E{X~J - E{X;} by Doob's Optional Sampling Theorem. Moreover E{X~J - E{X;} = E{[X,X]Sr - [X,X]r} = 0, by Corollary 3. Therefore E{(Xsr - X r )2} = 0, and XSr = X r a.s. Moreover this implies X q = XSq a.s. on {Sq = Sr} for each pair of rationals (r,q), and therefore we deduce that any interval of constancy of [X, X] is also one of X. 0 Note that the continuity of the local martingale X is essential in Corol- lary 5. Indeed, let Nt be a Poisson process, and let M t = Nt - t. Then M is a martingale and [M, Mlt = Nt; clearly M has no intervals of constancy while N is constant except for jumps. Theorem 28. Let X be a quadratic pure jump semimartingale. Then for any semimartingale Y we have [X, Y]t = XoYo+ L ~Xs~Ys' O 76 II Semimartingales and Stochastic Integrals Proof. First assume (without loss of generality) that X o = Yo = O. It suffices to establish the following result and then apply it again, by the symmetry of the form [".J, and by the asso- ciativity of the stochastic integral (Theorem 19). First suppose H is the indicator of a stochastic interval. That is, H = 1 [O,T], where T is a stopping time. Establishing (*) is equivalent in this case to showing [XT , Y] = [X, y]T, a result that is an obvious consequence of Theorem 23, which approximates [X, Y] by sums. Next suppose H = U1(s,Tj, where S, T are stopping times, S :s; T a.s., and U E Fs. Then JHsdXs = U(XT - XS), and by Theorem 23 [H· X, Y] = U{[XT ,Y] - [X s , Y]} = U{[X, Yf - [X, Y]s} = J Hsd[X, Y]s' The result now follows for H E 5 by linearity. Finally, suppose HElL and let Hn be a sequence in 5 converging in ucp to H. Let zn = Hn. X, Z = H· X. We know zn, Z are all semimartingales. We have JH';d[X, Y]s = [zn, YJ, since Hn E 5, and using integration by parts [zn, Y] = YZn - J Y_dZn - J Z~dY = YZn - JLHndX - JZ~dY. By the definition of the stochastic integral, we know zn ----t Z in ucp, and since Hn ----t H (ucp) , letting n ----t 00 we have lim [zn, Y] = YZ - JLHdX - JZ_dY n--->oo = Y Z - JY_dZ - JZ_dY = [Z, YJ, again by integration by parts. Since limn--->oo JH';d[X, Y]s = JHsd[X, Y]s, we have [Z, Y] = [H. X, Y] = JHsd[X, Y]s, and the proof is complete. 0 Example. Let E t be a standard Wiener process with Eo = 0, (i.e., Brownian motion). E; - t is a continuous martingale by Theorem 27 of Chap. 1. Let HElL be such that E{J~H;ds} < 00, each t 2: O. By Theorem 28 of Chap. I 6 The Quadratic Variation of a Semimartingale 77 we have [B, B]t = t, hence [H· B, H . B]t = J~ H;ds. By the martingale preservation property, JHsdBs is also a continuous local martingale, with (H· B)o = O. By Corollary 3 to Theorem 27 E{(l t HsdBs)2} = E{[H . B, H . B]tl = E{l t H;ds}. It was this last equality, that was crucial in K. Ito's original treatment of a stochastic integral. Theorem 30. Let H be a cadlag, adapted process, and let X, Y be two semimartingales. Let an be a sequence of random partitions tending to the identity. Then 2:,HTt(XT;'+l - XTt)(yT:+l _ yTin) converges in uc:p to JHs_d[X, y]s (Ho- = 0). Here an = (0 ::::; To ::::; Tf ::::; ... ::::;Tt::::; .. · ::::;TkJ· Proof. By the definition of quadratic variation, [X, Y] = XY -X_·Y - y_ ·X, where X_ . Y denotes the process (J~ Xs-dYsh~o. By the associativity of the stochastic integral (Theorem 19) H_· [X, Y] = H_· (XY) - H_ . (X_· Y) - H_· (Y_. X) = H_ . (XY) - (H_X_)· Y - (H_Y_)· X = H_ . (XY) - (HX)_ . Y - (HY)_ . X. By Theorem 21 the above is the limit of _ yTt (XT:'p _ XTin )} = 2:, HTt (XT,np - XTin)(yT:'p _ yTt). o 78 II Semimartingales and Stochastic Integrals 7 Ito's Formula (Change of Variables) Let A be a process with continuous paths of finite variation on compacts. If HElL we know by Theorem 17 that the stochastic integral H . A agrees a.s. with the path-by-path Lebesgue-Stieltjes integral f HsdAs. In Sect. 7 of Chap. I (Theorem 54) we proved the change of variables formula for fECI, namely We also saw at the end of Sect. 4 of this chapter that for a standard Wiener process B with Bo = 0, tIli o BsdBs = 2Bi - 2 t . Taking f(x) = x 2 /2, the above formula is equivalent to which does not agree with the Lebesgue-Stieltjes change of variables formula (*). In this section we will state and prove a change of variables formula valid for all semimartingales. We first mention, however, that the change of variables formula for con- tinuous Stieltjes integrals given in Theorem 54 of Chap. I has an extension to right continuous processes of finite variation on compacts. We state this result as a theorem but we do not prove it here because it is merely a special case of Theorem 32, thanks to Theorem 26. Theorem 31 (Change of Variables). Let V be an FV process with right continuous paths, and let f be such that l' exists and is continuous. Then (J(vt))t~O is an FV process and f(vt) - f(vo) = it f'(Vs-)dVs + I: {f(Vs) - f(Vs-) - f'(Vs-)~Vs}' 0+ O 7 Ito's Formula (Change of Variables) 79 f(Xt ) - f(Xo) = t f'(Xs-)dXs + ~ t J"(Xs_)d[X,X]~Jo+ Jo+ + l:: {f(Xs) - f(Xs-) - I'(Xs-)~Xs}' O 0, and let an be a refining sequence of random partitions5 of [0, t] tending to the identity [an = (0 = To::::; TI' ::::; ... ::::; T:: n = t)J. Then k n f(Xt ) - f(Xo) = L {f (XT'+l) - f (XT,n)} i=O = l:: l' (XT,n) (XT:+ 1 - XT,n) i +~l::J"(XT,n) (XTi+1-XT,nf + l::R(XT,n,XT'+l) i i 5 Note that it would suffice for this proof to restrict attention to deterministic partitions. 80 II Semimartingales and Stochastic Integrals The first sum converges in probability to the stochastic integral J~ !'(Xs- )dXs by Theorem 21; the second sum converges to ~ J~ f"(Xs)d[X, X]s in probabil- ity by Theorem 30. It remains to consider the third sum Ei R ( X Tr , XT:'tl)' But this sum is majorized, in absolute value, by supr(IXTr+l - XTr IHI:(XT:'tl - XTr )2}, • and since Li(XTr+l - XTr)2 converges in probability to [X, X]t (Theo- rem 22), the last term will tend to °if limn--->oo sUPi 1'(1 XTtj. 1 - XTr I) = 0. However s f-* Xs(w) is a continuous function on [0, t], each fixed w, and hence uniformly continuous. Since limn--->oo sUPi ITI.t l - Trl = °by hypoth- esis, we have the result. Thus, in the continuous case, f(Xt ) - f(Xo) = J~ !'(Xs-)dXs + ~ J~ f"(Xs-)d[X, X s], for each t, a.s. The continuity of the paths then permits us to remove the dependence of the null set on t, giving the complete result in the continuous case. Proof for the general case. X is now given as a right continuous semimartin- gale. Once again we have a representation as in (* * *), but we need a closer analysis. For any t > °we have EO ) - f(XTT»} = '"' {f(Xs) - f(Xs-)} ,n ~ 1.+1 1. L-J i,A sEA and by Taylor's formula L {f(XT:'tJ - f(XTin) } i,B = I: I'(XTin)(XT:'tl - X Tr ) + ~ I: I"(XT;, )(XT:'tl - X Tr )2 i i - l:: {I' (XTin) (XT:'tl - X Tr ) + ~1"(XTin) (XT:'tl - XTr )2 } i,A + LR(XTr,XT:'iJ. i,B 7 Ito's Formula (Change of Variables) 81 As in the continuous case, the first two sums on the right side above converge respectively to J~+ !'(Xs-)dXs and ~ J~+ f"(Xs-)d[X, X]s' The third sum converges to - I: {f'(Xs-)~Xs+ ~J"(Xs_)(~Xs)2}. sEA Assume temporarily that IXsl :::; k, some constant k, all s :::; t. Then f" is uniformly continuous, and using the right continuity of X we have lim sup I:R(XTt,XT;,+J:::; r(c+)[X,X]t, n i,B where r(c+) is limsuPo!e:r(J). Next let c tend to O. Then r(c+)[X,X]t tends to 0, and L {f(Xs) - f(Xs-) - f'(Xs-)f:.Xs - ~f"(Xs_)(f:.Xs)2} sEA(e:,t) tends to the series in (* * *), provided this series is absolutely convergent. Let Vk = inf{t > 0: IXtl:::: k}, withXo = O. By first establishing (***) for X1[oYk)' which is a semimartingale since it is the product of two semimartin- gales (Corollary 2 of Theorem 22), it suffices to consider semimartingales taking their values in intervals of the form [-k, k]. For f restricted to [-k, k] we have If(y) - f(x) - (y - x)f'(x)1 :::; C(y - X)2. Then I: If(Xs) - f(Xs-) - f'(Xs-)f:.Xsl :::; C I: (f:.Xs)2 :::; C[X,X]t < 00, O 82 II Semimartingales and Stochastic Integrals The stochastic integral calculus, as revealed by Theorems 32 and 33, is different from the classical Lebesgue-Stieltjes calculus. By restricting the class of integrands to semimartingales made left continuous (instead of L), one can define a stochastic integral that obeys the traditional rules of the Lebesgue- Stieltjes calculus. Definition. Let X, Y be semimartingales. Define the Fisk-Stratonovich integral of Y with respect to X, denoted J~ Ys- 0 dXs, by t Ys- 0 dXs == t Ys_dXs + ~[Y,X]~.io io 2 The Fisk-Stratonovich integral is often referred to as simply the Stratono- vich integral. The notation "0" is called Ito's circle. Note that we have defined the Fisk-Stratonovich integral in terms of the semimartingale integral. With some work one can slightly enlarge the domain of the definition and we do so in Sect. 5 of Chap. V. In particular, Theorem 34 below is proved with the weaker hypothesis that f E C2 (Theorem 20 of Chap. V). We will write the F-S integral as an abbreviation for the Fisk-Stratonovich integral. Theorem 34. Let X be a semimartingale and let f be C3 . Then f(Xt )- f(Xo) = it f'(Xs-)odXs+ L {f(Xs) - f(Xs-) - f'(Xs-)~Xs}. 0+ O 7 ItO's Formula (Change of Variables) 83 fact that [X, X] has paths of finite variation, to be Lo 84 II Semimartingales and Stochastic Integrals Theorem 36. Let X, Y be cadlag semimartingales, let Zt = X t + iyt, and let f be analytic. Then f(Zt) = f(Zo) + r t J'(Zs-)dZs + ~ t f"(Zs-)d[Z,Z]~Jo+ 2 Jo+ + L {J(Zs) - f(Zs-) - J'(Zs-)~Zs}' O 8 Applications of Ito's Formula 85 Vi = IT (1 + ~Xsl{I~Xslo. Since >.B has no jumps we have 86 II Semimartingales and Stochastic Integrals A2 A2 £(AB)t = exp{ABt - 2[B, B]t} = exp{ABt - 2 t }. Moreover, since £(AB}t = 1 + A f~ £(AB)s_dBs we see that £(AB)t eAB, - >'22 t is a continuous martingale. The process £ (AB) is sometimes referred to as geometric Brownian motion. Note that the previous theorem gives us £(X) in closed form. We also have the following pretty result. Theorem 38. Let X and Y be two semimartingales with X o = Yo = O. Then £(X)£(Y) = £(X + Y + [X, YJ). Proof Let Ut = £(X)t and vt = £(Yk Then the integration by parts formula gives that UtVi - 1 = f~+ Us_dVs + f~+ Vs_dUs + [U, VJt. Since U and V are exponentials, this is equivalent to Letting W t = Utvt, we deduce that W t = 1 + f~Ws_d(X + Y + [X, Y])s so that W = £(X + Y + [X, Y]), which was to be shown. 0 Corollary. Let X be a continuous semimartingale, X o = O. Then £(X)-l = £(-X + [X, XJ). Proof By Theorem 38, £(X)£(-X + [X, X]) = £(X + (-X + [X, X]) + [X, -X]), since [-X, [X, XJJ = O. However £(0) = 1, and we are done. o In Sect. 9 of Chap. V we consider general linear equations. In particular, we obtain an explicit formula for the solution of the equation where Z is a continuous semimartingale. We also consider more general in- verses of stochastic exponentials. See, for example, Theorem 63 of Chap. V. Another application of the change of variables theorem (and indeed of the stochastic exponential) is a proof of Levy's characterization of Brownian motion in terms of its quadratic variation. Theorem 39 (Levy's Theorem). A stochastic process X = (Xth~o is a standard Brownian motion if and only if it is a continuous local martingale with [X,X]t = t. 8 Applications of Ito's Formula 87 Proof We have already observed that a Brownian motion B is a continuous local martingale and that [B, B]t = t (see the remark following Theorem 22). Thus it remains to show sufficiency. Fix u E IR and set F(x, t) = exp{iux + ~2 t}. Let Zt = F(Xt , t) = exp{iuXt + (u22 ) t}. Since F E C2 we can apply Ito's formula (Theorem 33) to obtain Zt = 1 + iu it ZsdXs + ~2 it Zsds _ ~2 it Zsd[X, X]s = 1 + iu it ZsdXs, which is the exponential equation. Since X is a continuous local martingale, we now have that Z is also one (complex-valued, of course) by the martingale preservation property. Moreover stopping Z at a fixed time to, zto , we have that zto is bounded and hence a martingale. It then follows for 0 ::; s < t that E{exp{iu(Xt - Xs)}IFs} = exp { - ~2 (t - S)}. Since this holds for any u E IR we conclude that X t - X s is independent of Fs and that it is normally distributed with mean zero and variance (t - s). Therefore X is a Brownian motion. 0 Observe that if M and N are two continuous martingales such that M N is a martingale, then [M, N] = 0 by Corollary 2 of Theorem 27. Therefore if B t = (Bl, ... ,Bf) is an n-dimensional standard Brownian motion, B:Bf is a martingale for i =I- j, and we have that if i = j, if i =I- j. Theorem 39 then has a multi-dimensional version, which has an equally simple proof. Theorem 40 (Levy's Theorem: Multi-dimensional Version). Let X = (Xl, ... ,Xn) be continuous local martingales such that ifi = j, if i =I- j. Then X is a standard n-dimensional Brownian motion. As another application ofIto's formula, we exhibit the relationship between harmonic and subharmonic functions and martingales. Theorem 41. Let X = (Xl, ... , xn) be an n-dimensional continuous local martingale with values in an open subset D ofIRn. Suppose that [Xi,Xj] = 0 ifi =I- j, and [Xi, Xi] = A, 1 ::; i::; n. Let u: D -+ IR be harmonic (resp. subharmonic). Then u(X) is a local martingale (resp. submartingale). 88 II Semimartingales and Stochastic Integrals Proof. By Ito's formula (Theorem 33) we have u(Xt} - u(Xo) = t Vu(Xs) . dXs + ~ t .6.u(Xs)dAsJo+ 2 Jo+ where the "dot" denotes Euclidean inner product, V denotes the gradient, and .6. denotes the Laplacian. If u is harmonic (subharmonic), then .6.u = O(.6.u ?': 0) and the result follows. 0 If B is a standard n-dimensional Brownian motion, then B satisfies the hypotheses of Theorem 41 with the process At = t. That u(Bt ) is a submartin- gale (resp. supermartingale) when u is subharmonic (resp. superharmonic) is the motivation for the terminology submartingale and supermartingale. The relationship between stochastic calculus and potential theory suggested by Theorem 41 has proven fruitful (see, for example, Doob [56]). Levy's characterization of Brownian motion (Theorem 39) allows us to prove a useful change of time result. Theorem 42. Let M = (Mth?-o be a continuous local martingale with Mo = 0 and such that limt--+oo[M, MJt = 00 a.s. Let Ts = inf{t > 0: [M,MJt > s}. Define 9s = Frs and B s = Mrs' Then (Bs,9s)s?-0 is a standard Brownian motion. Moreover ([M, Mlt)t?-o are stopping times for (9s)s?-o and Mt = B[M,MJ, a.s. 0:::::; t < 00. That is, M can be represented as a time change of a Brownian motion. Proof The (Ts)s?-o are stopping times by Theorem 3 of Chap. I. Each Ts is finite a.s. by the hypothesis that limt--+oo [M, Mlt = 00 a.s. Therefore the (I-fields 9s = Frs are well-defined. The filtration (9s)s?-o need not be right continuous, but one can take 1{s = 9s+ = Fr s + to obtain one. Note further that HM, M]t :::::; s} = {Ts ?': t}, hence ([M, M]t}t?-o are stopping times for the filtration 9 = (9s)s>o, By Corollary 3 of Theorem 27 we have E{Mf,s} = E{[M, M]r s } = s < 00, since [M, M]rs = s identically because [M, M] is continuous. Thus the time changed process is square integrable. Moreover by the Optional Sampling Theorem. Also E{B~ - B;19s} = E{(Bu - B s)219s} = E{(Mr u - Mr.)2IFrJ = E{[M,M]ru - [M,M]rsIFrs } = u- s. 8 Applications of Ito's Formula 89 Therefore B; - s is a martingale, whence [B, B]s = s provided B has contin- uous paths, by Corollary 2 to Theorem 27. We want to show that B s = Mrs has continuous paths. However by Corol- lary 5 of Theorem 27 almost surely all intervals of constancy of [M, M] are also intervals of constancy of M. It follows easily that B is continuous. It remains to show that M t = B[M,MJ,. Since B s = Mrs' we have that B[M,MJ, = Mr[M,Mlt' a.s. Since (Ts)sc.o is the right continuous inverse of [M, M], we have that T[M,MJ, ?': t, with equality holding if and only if t is a point of right in- crease of [M, M]. (If (Ts)sc.o were continuous, then we would always have that T[M,MJ, = t.) However T[M,MJ, > t implies that t f-+ [M, M]t is constant on the interval (t, T[M,MJ,); thus by Corollary 5 of Theorem 27 we conclude M is constant on (t, T[M,Ml,)· Therefore B[M,Ml' = Mr[M,Mlt = M t a.s., and we are done. 0 Another application of the change of variables formula is the determination of the distribution of Levy's stochastic area process. Let B t = (Xt , yt) be an JR2-valued Brownian motion with (Xo,Yo) = (0,0). Then during the times s to s +ds the chord from the origin to B sweeps out a triangular region of area ~RsdNs, where R = VX2+y2s s s and Ys X sdNs = - R s dXs + R s dYs· Therefore the integral At = J~ RsdNs = J~(-YsdXs+Xsd~) is equal to twice the area swept out from time 0 until time t. Paul Levy found the characteristic function of At and therefore determined its distribution. Theorem 43 is known as Levy's stochastic area formula. Theorem 43. Let B t = (Xt , yt) be an JR2-valued Brownian motion, B o (0,0), u E lR. Let At = J~ Xsd~ - J~ YsdXs. Then . A 1E{eW '}- O:::::;t 90 II Semimartingales and Stochastic Integrals Next observe that, from the above calculation, d[V, Vlt = (-iu}t - a(t)Xt )2dt + (iuXt - a(t)}t)2dt = (a(t)2 - u2)(X; + ~2)dt. Using the change of variables formula and the preceding calculations 1dev, = eV'(dvt + 2"d[V, VJd = eV' (-iu}t - a(t)Xt)dXt + eV'(iuXt - a(t)}t)d}t + ~eV'dt{(a(t)2 - u2 - a'(t))(X; + ~2) + 2fJ'(t) - 2a(t)}. Therefore ev, is a local martingale provided a'(t) = a(t)2 - u2 fJ'(t) = a(t). Next we fix to > 0 and solve the above ordinary differential equations with a(to) = fJ(to) = O. The solution is a(t) = u tanh(u(to - t)) fJ(t) = -log cosh(u(to - t)), where tanh and cosh are hyperbolic tangent and hyperbolic cosine, respec- tively. Note that for 0 ::::; t ::::; to, Thus ev,, 0 ::::; t ::::; to is bounded and is therefore a true martingale, not just a local martingale, by Theorem 51 of Chap. I. Therefore, However, vto = iuAto since a(to) = fJ(to) = o. As Ao = X o = Yo = 0, it follows that Vo = -log cosh(uto). We conclude that E { eiuA,o } = exp{ -log cosh(uto)} 1 cosh(uto) , and the proof is complete. o There are of course other proofs of Levy's stochastic area formula (e.g., Yor [246], or Levy [150]). As a corollary to Theorem 43 we obtain the density for the distribution of At. 8 Applications of Ito's Formula 91 Corollary. Let B t = (Xt , yt) be an ]R2-valued Brownian motion, B o = (0,0), and set At = J~ Xsd~ - J~ YsdXs. Then the density function for the distri- bution of At is 1fA (x) - -00 < x < 00. t - 2t cosh(1fx/2t)' Proof By Theorem 43, the Fourier transform (or characteristic function) of At is E{e iuAt } = cos~(ut)· Thus we need only to calculate 2~ J c~::(:t) duo The integrand is of the form f(z) = ~i~j = c~::;:t). Since cosh(zt) has a pole at in hZo = 2t' we ave P(zo) enx/ 2t Res(f,zo) = -Q( ) = -.-. , Zo zt Next we integrate along the closed curve Cr traversed counter clockwise, and given by y = 0, -r ::; x ::; r, C1r 1f C2x = r, 0< Y 92 II Semimartingales and Stochastic Integrals i 100 e-iuxlim f(z)dz = (1 + e7rx / t ) ( ) dur--+oo C r -00 cosh ut = 21fi Res(f, zo) e7rx / 2t = 21f--. t Finally we can conclude 1 100 e- iux 1 ( 21feTt- ) 21f _oocosh(ut)du= 21f t(l+e7) 1 t(e ;: + e -2~x ) 1 2t cosh( ;~ ) . o The stochastic area process A shares some of the properties of Brownian motion, as is seen by recalling that At = J~ R s dNs , where N is a Brownian motion by Levy's Theorem (Theorem 39), and Nand R are independent (this must be proven, of course). For example A satisfies a reflection principle. If one changes the sign of the increments of A after a stopping time, the process obtained thereby has the same distribution as that of A. One can use this fact to show, for example, that if St = sUPO Bibliographic Notes 93 1940, but was killed later that year in action in the Second World War, and thus his results were lost until recently [26, 27]. Doob stressed the martingale nature of the Ito integral in his book [55] and proposed a general martingale integral. Doob's proposed development depended on a decomposition theo- rem (the Doob-Meyer decomposition, Theorem 8 of Chap. III) which did not yet exist. Meyer proved this decomposition theorem in [163, 164], and com- mented that a theory of stochastic integration was now possible. This was begun by Courrege [36], and extended by Kunita and Watanabe [134], who revealed an elegant structure of square integrable martingales and established a general change of variables formula. Meyer [166, 167, 168, 169] extended Kunita and Watanabe's work, realizing that the restriction of integrands to predictable processes is essential. He also extended the integrals to local mar- tingales, which had been introduced earlier by Ito and Watanabe [102]. Up to this point, stochastic integration was tied indirectly to Markov processes, by the assumption that the underlying filtration of O"-algebras be "quasi-left continuous." This hypothesis was removed by Doleans-Dade and Meyer [53], thereby making stochastic integration a purely martingale theory. It was also in this article that semimartingales were first proposed in the form we refer to as classical semimartingales in Chap. III. A different theory of stochastic integration was developed independently by McShane [154, 155], which was close in spirit to the approach given here. However it was technically complicated and not very general. It was shown in Protter [200] (building on the work of Pop-Stojanovic [194]) that the theory of McShane could for practical purposes be viewed as a special case of the semimartingale theory. The subject of stochastic integration essentially lay dormant for six years until Meyer [171] published a seminal "course" on stochastic integration. It was here that the importance of semimartingales was made clear, but it was not until the late 1970's that the theorem of Bichteler [13, 14], and Del- lacherie [42] gave an a posteriori justification of semimartingales. The seem- ingly ad hoc definition of a semimartingale as a process having a decomposi- tion into the sum of a local martingale and an FV process was shown to be the most general reasonable stochastic differential possible. (See also Kuss- maul [138] in this regard, and the bibliographic notes in Chap. III.) Most of the results of this chapter can be found in Meyer [171], though they are proven for classical semimartingales and hence of necessity the proofs are much more complicated. Theorem 4 (Stricker's Theorem) is (of course) due to Stricker [218]; see also Meyer [173]. Theorem 5 is due to Meyer [175]. There are many other methods of expanding a filtration and still preserving the semimartingale property. For further details, see Chap. VI on expansion of filtrations. Theorem 14 is originally due to Lenglart [142]. Theorem 16 is known to be true only in the case of integrands in JL. The local behavior of the integral (Theorems 17 and 18) is due to Meyer [171] (see also McShane [154]). The a.s. 94 Exercises for Chapter II Kunita-Watanabe inequality, Theorem 25, is due to Meyer [171], while the ex- pected version (the corollary to Theorem 25) is due to Kunita-Watanabe [134J. That continuous martingales have paths of infinite variation or are con- stant a.s. was first published by Fisk [73J (Corollary 1 of Theorem 27). The proof given here of Corollary 4 of Theorem 27 (that a continuous local martin- gale X and its quadratic variation [X, X] have the same intervals of constancy) is due to Maisonneuve [151]. The proof of Ito's formula (Theorem 32) is by now classic; however we benefited from Follmer's presentation of it [75]. A popular alternative proof which basically bootstraps up form the formula for integration by parts, can be found (for example) on pages 57-58 of J. Jacod and A. N. Shiryaev [110]. The Fisk-Stratonovich integral was developed independently by Fisk [73] and Stratonovich [217], and it was extended to general semimartingales by Meyer [171]. Theorem 35 is inspired by the work of Getoor and Sharpe [215]. The stochastic exponential of Theorem 37 is due to DoU~ans-Dade [49J. It has become extraordinarily important. See, for example, Jacod-Shiryaev [110]. The pretty formula of Theorem 38 is due to Yor [237]. Exponentials have of course a long history in analysis. For an insightful discussion of exponentials see Gill-Johansen [82]. That every continuous local martingale is the time change of a Brownian motion is originally due to Dubins-Schwarz [59] and Dambis [37]. The proof of Levy's stochastic area formula (Theorem 43) is new and is due to Janson. See Janson-Wichura [113] for related results. The original result is in Levy [150], and another proof can be found in Yor [246J. Exercises for Chapter II Exercise 1. Let B be standard Brownian motion and let f be a function mapping IR -+ IR which is continuous except for one point where there is a jump discontinuity. Show that X t = f(Bt ) cannot be a semimartingale. Exercise 2. Let Q « P (Q and P are both probability measures and Q is absolutely continuous with respect to P). Show that if X n -+ X in P- probability, then X n -+ X in Q-probability. Exercise 3. Let (X, Y) be standard two dimensional Brownian motion. (That is, X and Yare independent one dimensional standard Brownian motions.) Let a < a < 1. Set B t = aXt + ~yt. Show that B is a standard one dimensional Brownian motion. Compute [X, B] and [Y,B]. Exercise 4. Let f : IR+ -+ IR+ be non-decreasing and continuous. Show that there exists a continuous martingale M such that [M, M]t = f(t). Exercise 5. Let B be standard Brownian motion and let H E JL be such that IHtl = 1, all t. Let Mt = J~ HsdBs. Show that M is also a Brownian motion. Exercises for Chapter II 95 Exercise 6. Give an example of semimartingales xn, X, Y such that limn--+oo XI' = X t a.s., all t, but limn--+oo [xn, Ylt =I- [X, Ylt a.s., for some t. Exercise 7. Let Y, Z be semimartingales, and let Hn, H E JL. Suppose limn--+oo Hn = H in ucp. Let X n = JH;dYs and X = JHsdYs. Show that limn--+oo [X n , Zlt = [X, Zlt in ucp, all t. Exercise 8. Suppose I and all In are C2 , and that In and I~ converge uniformly on compacts to I and f', respectively. Let X, Y be continuous semimartingales and show that limn--+oo[fn(X), Y] = [/(X), Yl. Exercise 9. Let M be a continuous local martingale and let A be a continu- ous, adapted process with paths of finite variation on compact time sets, with Ao = O. Let X = M + A. Show that [X, Xl = [M, Ml a.s. Exercise 10. Let X be a semimartingale and let Q be another probability law, with Q < < P. Let [X, XlP denote the quadratic variation of X considered as a P semimartingale. Show that [X,XlQ = [X,Xl P , Q-a.s. Exercise 11. Let B be standard Brownian motion, and let T = inf{ t > 0 : B t = 1 or B t = - 2}. (a) Show that T is a stopping time. (b) Let M = Bt/\T and let N = -M. Show that M and N are continuous martingales and that [M, Mlt = [N, Nlt = t 1\ T. Show that nevertheless M and N have different distributions. Exercise 12. Let X be a semimartinagle such that L::o JHsdXs in ucp. *Exercise 14. Let X be a semimartingale, and let A be a non-negative, continuous, increasing process with Aoo = 00 a.s. Suppose limt--+oo J~(l + As)-ldXs exists and is finite a.s. Show that limt--+oo 1;- = 0 a.s. Exercise 15. Let M be a continuous local martingale and assume [M, Mloo = 00 a.s. Show that limt--+oo [M~k]t = 0 a.s. Exercise 16. Let B be a standard Brownian motion and let H be adapted and continuous. Show that for fixed t, with convergence in probability. 96 Exercises for Chapter II Exercise 17. Let Y be a continuous, adapted process with Yo = 0 and with Y constant on [1,(0). Let Show that X n is a semimartingale, each n, but that limn - HXJ X n = Y need not be a semimartingale. Exercise 18. (Continuation of Exercise 17.) Let B be a standard Brownian motion and let X n = n JL* B s dsl{t>*}. Solve dZ;- = Z~dX~, Zr; = 1, and show that limn --+oo Xf = B t a.s., each t, but that limn --+oo zn =I- Z, where Z solves the equation dZt = ZtdBt. Exercise 19. (Related to Exercises 17 and 18.) Let Af = ~sin(nt), 0:::; t:::; 7f/2. Show that An is a semimartingale for each n, and that Jo~ IdA~1 = 1, each n (this means the total variation process, path-by-path, of A), but that lim sup IArl = o. n--+oo o ]R be given by u(y) = Ilyil-I. Assume Bo = x a.s., where x =I- 0, show the following. (a) Mt = u(Bt ) is a local martingale. (b) limt--+oo EX(u(Bt )) = O. (c) u(Bd E L 2 , each t 2: O. (d) SUPt~O EX(u(Bd 2 ) < 00. Conclude that u(B) is not a martingale. Exercise 21. Let A, C be two non-decreasing cadla,g processes with Ao Co = O. Assume that both Aoo = limt--+oo At and Coo = limt--+oo Ct are finite. (a) Show that AooCoo = JoOO(Aoo - As)dCs + JoOO(Coo - Cs-)dAs. (b) Deduce from part (a) the general formula Note that the above formulae are not symmetric. See also Theorem 3 in Chap. VI. Exercises for Chapter II 97 Exercise 22. Let A be non-decreasing and continuous, and assume A:>o = limt-HXl At is finite, Ao = O. Show that for integer p > 0, *Exercise 23 (expansion of filtrations). Let B be a standard Brownian motion on a filtered probability space (0, F, IF, P), and define a new filtration G by g2 = Ft V a(BI) and gt = nu>t g~. (a) Show that E{Bt - Bslgs} = ~=~(Bl - B s) for 0:::; s:::; t:::; 1. (b) Show that i t f\l Bl - B sBt = f3t + dso 1- s where f3 is a G-Brownian motion. (Note in particular that the process (Bt - J~ Bi=~'ds)oSt 0 and T = inf{t > 0: Bt E (-a,b)C}. (a) Show that 1 2 [ b-a]Xt = exp {2e t}cos e(Bt - -2-) is a local martingale. (b) Show that 98 Exercises for Chapter II (c) For 0 ::; e< a~b' show that XT is a positive supermartingale, and deduce a+b 1 a-b cos(-2-e)E{exp{"2e2T}}::; cos(-2-e ). (d) Use (c) to show that XT = sUPso be standard Brownian motion, and let T = inf{t > 0: B t E (-d,d)C}. L;t M = B T . Show that (a) if d < ~, then E{expH[M,M]T}} < 00; but (b) if d = ~, then E{expH[M, M]T}} = 00. (Hint: Use Exercise 25.) Exercise 27. Let (Bt , Ft)t>o be standard Brownian motion, and let X t = e-Qt(Xo+a J; eQSdBs ). Sho~ that X is a solution to the stochastic differential equation dXt = -aXtdt + adBt . Exercise 28. Let B be a standard Brownian motion and let £(B) denote the stochastic exponential of B. Show that limt->oo £(B)t = 0 a.s. Exercise 29. Let X be a semimartingale. Show that d(_l_) = -dX + d[X,X] £(X) £(X) Exercise 30. Let B be a standard Brownian motion. (a) Show that M is a local martingale, where (b) Calculate [M, M]t, and show that M is a martingale. (c) Calculate E{eBt }. The next eight problems involve a topic known as changes of time. For these problems, let (fl,F,IF,P) satisfy the usual hypotheses. A change of time R = (Rt)t>o is a family of stopping times such that for every W E fl, the function R.(w) is non-decreasing, right continuous, Rt < 00 a.s., and R o = O. Let Qt = FRt • Change of time is discussed further in Sect. 3 of Chap. IV. Exercise 31. Show that G = (Qtk::o satisfies the usual hypotheses. Exercise 32. Show that if M is an IF uniformly integrable martingale and M t := MRt' then M is a G martingale. Exercises for Chapter II 99 Exercise 33. If M is an IF (right continuous) local martingale, show that M is a G semimartingale. *Exercise 34. Construct an example where M is an IF local martingale, but M is not a G local martingale. (Hint: Let (Xn)nEN be an adapted process. It is a local martingale if and only if IXnldP is a a-finite measure on Fn- 1 , and E{XnIFn-Il = Xn- 1 , each n ~ 1. Find X n where IXnldP is not a-finite on Fn - 2 , any n, and let Rn = 2n.) *Exercise 35. Let R be a time change, with s f-+ R s continuous, strictly increasing, Ro = 0, and Rt < 00, each t 2 O. Show that for a continuous semimartingale X, for bounded H E IT.... *Exercise 36. Let R and X be as in Exercise 35. No longer assume that Rt < 00 a.s., each t ~ 0, but instead assume that X is a finite variation process. Let At = inf{s > 0: Rs > t}. (a) Show that R strictly increasing implies that A is continuous. (b) Show that R continuous implies that A is strictly increasing. (c) Show that for general R, RAt ~ t, and if R is strictly increasing and continuous then RAt = t. (d) Show that for bounded HElL we have (e) Show that for bounded HElL we have l Rt ltl\AooHsdAs = HRsds.Ro 0 See in this regard Lebesgue's change of time formula, given in Theorem 45 of Chap. IV. *Exercise 37. Let R be a change of time and let G be the filtration given by gt = FRt ' Let At = inf{s > 0 : R s > t}. Show that A = (At)t>o is a change of time for the filtration G. Show also that if t -+ Rt is continu~us a.s., Ro = 0, and Roo = 00, then RAt = t a.s., t ~ O. *Exercise 38. Let A, G, be as in Exercise 37 and suppose that RAt = t a.s., t 2 O. Show that gAt eFt, each t ~ O. *Exercise 39. A function is Holder continuous of order 0: if If(x)- f(y)1 ::; Klx-yl". Show that the paths of a standard Brownian motion are a.s. nowhere locally Holder continuous of order 0: for any 0: > 1/2. (Hint: Use the fact that limn->oo L7rn[O,tJ(Bti+1 - BtJ2 = t.) III Semimartingales and Decomposable Processes 1 Introduction In Chap. II we defined a semimartingale as a good integrator and we developed a theory of stochastic integration for integrands in IL, the space of adapted processes with left continuous, right-limited paths. Such a space of integrands suffices to establish a change of variables formula (or "Ito's formula"), and it also suffices for many applications, such as the study of stochastic differential equations. Nevertheless the space IL is not general enough for the consider- ation of such important topics as local times and martingale representation theorems. We need a space of integrands analogous to measurable functions in the theory of Lebesgue integration. Thus defining an integral as a limit of sums~whichrequires a degree of smoothness on the sample paths~is inade- quate. In this chapter we lay the groundwork necessary for an extension of our space of integrands, and the stochastic integral is then extended in Chap. IV. Historically the stochastic integral was first proposed for Brownian motion, then for continuous martingales, then for square integrable martingales, and finally for processes which can be written as the sum of a locally square integrable local martingale and an adapted, cadlag processes with paths of finite variation on compacts; that is, a decomposable process. Later Doleans- Dade and Meyer [53] showed that the local square integrability hypothesis could be removed, which led to the traditional definition of a semimartingale (what we call a classical semimartingale). More formally, let us recall two definitions from Chaps. I and II and then define classical semimartingales. Definition. An adapted, cadlag process A is a finite variation process (FV) if almost surely the paths of A are of finite variation on each compact interval of [0,00). We write Iooo IdAs I or IAloo for the random variable which is the total variation of the paths of A. Definition. An adapted, cadlag process X is decomposable if there exist processes N, A such that 102 III Semimartingales and Decomposable Processes with No = A o = 0, N a locally square integrable local martingale, and A an FV process. Definition. An adapted, cadlag process Y is a classical semimartingale if there exist processes N, B with No = Bo = °such that where N is a local martingale and B is an FV process. Clearly an FV process is decomposable, and both FV processes and de- composable processes are semimartingales (Theorems 7 and 9 of Chap. II). The goal of this chapter is to show that a process X is a classical semimartin- gale if and only if it is a semimartingale. To do this we have to develop a small amount of "the general theory of processes." The key result is Theorem 25 which states that any local martingale M can be written where N is a local martingale with bounded jumps (and hence locally square integrable), and A is an FV process. An immediate consequence is that a classical semimartingale is decomposable and hence a semimartingale by The- orem 9 of Chap. II. The theorem of Bichteler and Dellacherie (Theorem 43) gives the converse: a semimartingale is decomposable. We summarize the results of this chapter, that are important to our treat- ment, in Theorems 1 and 2 which follow. Theorem 1. Let X be an adapted, cadlag process. The following are equiva- lent: (i) X is a semimartingale; (ii) X is decomposable; (iii) given 13 > 0, there exist M, A with Mo = Ao = 0, M a local martingale with jumps bounded by 13, A an FV process, such thatXt = Xo+Mt+At; (iv) X is a classical semimartingale. Definition. The predictable a-algebra P on JR+ x n is the smallest a- algebra making all processes in lL measurable. We also let P (resp. bP) denote the processes (resp. bounded processes) that are predictably measurable. The next definition is not used in this chapter, except in the Exercises, but it is natural to include it with the definition of the predictable a-algebra. Definition. The optional a-algebra 0 on JR+ x n is the smallest a-algebra making all cadlag, adapted processes measurable. We also let 0 (resp. bO) denote the processes (resp. bounded processes) that are optional. 2 The Classification of Stopping Times 103 Theorem 2. Let X be a semimartingale. If X has a decomposition X t = X o + Mt + At with M a local martingale and A a predictably measurable FV process, Mo = A o = 0, then such a decomposition is unique. In Theorem 1, clearly (ii) or (iii) each imply (iv), and (iii) implies (ii), and (ii) implies (i). That (iv) implies (iii) is an immediate consequence of the Fundamental Theorem of Local Martingales (Theorem 25). While Theo- rem 25 (and Theorems 3 and 22) is quite deep, nevertheless the heart of Theo- rem 1 is the implication (i) implies (ii), essentially the theorem of K. Bichteler and C. Dellacherie, which itself uses the Doob-Meyer decomposition theorem, Rao's Theorem on quasimartingales, and the Girsanov-Meyer Theorem on changes of probability laws. Theorem 2 is essentially Theorem 30. We have tried to present this succession of deep theorems in the most direct and elementary manner possible. In the first edition we were of the opinion that Meyer's original use of natural processes was simpler than the now universally accepted use of predictability. However, since the first edition, R. Bass has published an elementary proof of the key Doob-Meyer decompo- sition theorem which makes such an approach truly obsolete. We are pleased to use Bass' approach here; see [11]. 2 The Classification of Stopping Times We begin by defining three types of stopping times. The important ones are predictable times and totally inaccessible times. Definition. A stopping time T is predictable if there exists a sequence of stopping times (Tn)n~l such that Tn is increasing, Tn < T on {T > O}, all n, and limn->oo Tn = T a.s. Such a sequence (Tn) is said to announce T. If X is a continuous, adapted process with X o = 0, and T = inf{t : IXtl :2: c}, for some c > 0, then T is predictable. Indeed, the sequence Tn = inf{t : IXt I :2: c - ~} /\ n is an announcing sequence. Fixed times are also predictable. Definition. A stopping time T is accessible if there exists a sequence (Tk)k~l of predictable times such that 00 P( U{w: n(w) = T(w) < oo}) = P(T < 00). k=l Such a sequence (Tk)k~l is said to envelop T. Any stopping time that takes on a countable number of values is clearly ac- cessible. The first jump time of a Poisson process is not an accessible stopping time (indeed, any jump time of a Levy process is not accessible). Definition. A stopping time T is totally inaccessible if for every pre- dictable stopping time S, 104 III Semimartingales and Decomposable Processes P{w: T(w) = S(w) < oo} = O. Let T be a stopping time and A E :FT. We define TA(W) = {T(W), 00, if wE A, if w 1. A. It is simple to check that since A E :FT, TA is a stopping time. Note further that T = min(TA, TAC) = TA /\ TAco A simple but useful concept is that of the graph of a stopping time. Definition. Let T be a stopping time. The graph of the stopping time T is the subset ofJR+ x n given by ((t,w): 0::; t = T(w) < oo}; the graph of T is denoted by [T]. Theorem 3. Let T be a stopping time. There exist disjoint events A, B such that A U B = {T < oo} a.s., TA is accessible and TB is totally inaccessible, and T = TA /\ TB a.s. Such a decomposition is a.s. unique. Proof If T is totally inaccessible there is nothing to show. So without loss of generality we assume it is not. We proceed with an inductive construction: Let R 1 = T and take (}:1 = sup{P(S = R 1 < 00) : S is predictable}. Choose SI predictable such that P(SI = R 1 < 00) > -T and set VI = SI· Define R 2 = R1{V1#Rd. For the inductive step let (}:i = sup{P(S = R i < 00) : S is predictable} If (}:i = 0 we stop the induction. Otherwise, choose Si predictable such that P(Si = R i < 00) > T and set Vi = Si{Si does not equal any OfVj,lSjSi-l}O Let R i +1 = Ri{Vi#R;}. The graphs of the Vi are disjoint by construction (the finite valued parts, of course). Moreover the sets {RH1 =I- Ri, Ri < oo} are disjoint. Thus L ~i ::; L P(RH1 =I- R i , R i < 00) ::; 1, which implies that (}:i ~ o. The Ri form a non-decreasing sequence, so set U = limi-HXl Ri . If U is not totally inaccessible, there exists a predictable time W such that P(W = U) > 0 and hence P(W = U) > (}:i for some i. This contradicts how we chose Si at step i of the induction. Therefore U is totally inaccessible, and B = {U = T < oo}. 0 A beautiful application of Theorem 3 is Meyer's Theorem on the jumps of Markov processes, a special case of which we give here without proof. (We use the notation established in Chap. I, Sect. 5 and write IFIt = (:Fi)oStSoo.) Theorem 4 (Meyer's Theorem). Let X be a (strong) Markov Feller process for the probability pit, where the distribution of X o is given by p" and with its natural completed filtration IFIt. Let T be a stopping time with PIt(T > 0) = 1. Let A = {w : XT(W) =I- XT-(w) and T(w) < oo}. Then T = TA /\ TAc, where TA is totally inaccessible and TAo is predictable. 3 The Doob-Meyer Decompositions 105 A consequence of Meyer's Theorem is that the jump times of a Poisson process (or more generally a Levy process) are all totally inaccessible. We will need a small refinement of the concept of a stopping time a-algebra. This is particularly important when the stopping time is predictable. Definition. Let T be a stopping time. The a-algebra FT- is the smallest a-algebra containing Fo and all sets of the form An {t < T}, t > 0 and A EFt. Observe that FT- eFT, and also the stopping time T is FT- measurable. We also have the following elementary but intuitive result. We leave the proof to the reader. Theorem 5. Let T be a predictable stopping time and let (Tn)n;:::l be an announcing sequence for T. Then FT- = a{Un;:::1 FTn } = Vn;:::l FTn • 3 The Doob-Meyer Decompositions We begin with a definition. Let N denote the natural numbers. Definition. An adapted, cadlag process X is a potential if it is a non- negative supermartingale such that limt-HXl E{Xt } = o. A process (Xn)nEN is also called a potential if it is a nonnegative supermartingale for Nand limn->oo E{Xn} = o. Theorem 6 (Doob Decomposition). A potential (Xn)nEN has a decom- position X n = Mn - An, where An+l :::: An a.s., Ao = 0, An E Fn-l, and Mn = E{AooIFn }. Such a decomposition is unique. Proof Let Mo = X o and A o = o. Define M l = Mo + (Xl - E{XIIFo}), and Al = Xo - E{XIIFo}. Define Mn, An inductively as follows: Mn = Mn- l + (Xn - E{XnIFn-d), An = An- l + (Xn- l - E{XnIFn-tl)· Note that E{An } = E{Xo} - E{Xn } ::; E{Xo} < 00, as is easily checked by induction. It is then simple to check that Mn and An SO defined satisfy the hypotheses. Next suppose X n = Nn - Bn is another such representation. Then Mn - Nn = An - Bn and in particular M I - N I = Al - B I E Fo. Thus M l - N I = E{MI - N1IFo} = M o - No = X o - X o = 0, hence M l = N l . Continuing inductively shows M n = Nn , all n. 0 We wish to extend Theorem 6 to continuous time supermartingales. Note the unusual measurability condition that An E Fn- l which of course is stronger than simply being adapted. The continuous time analog is that the process A be natural or what turns out to be equivalent, predictably measur- able. 106 III Semimartingales and Decomposable Processes Throughout this paragraph we assume given a filtered probability space (0, F, IF, P) satisfying the usual hypotheses. Before we state the first decom- position theorem in continuous time, we establish a simple lemma. Lemma. Let (Yn)n;::::l be a sequence of random variables converging to 0 in L 2 . Then SUPt E{YnIFt}-+ 0 in L 2 . Proof Let Mr = E{Yn1Ft } which is of course a martingale for each fixed n. Using Doob's maximal quadratic inequality, E{suPt(Mr)2} ::; 4E{(M~Y} = 4E{Y,n -+ O. 0 Definition. We will say a cadlag supermartingale Z with Zo = 0 is of Class D if the collection {ZT : T a finite valued stopping time} is uniformly integrable. The name "Class D" was given by P. A. Meyer in 1963. Presumably he expected it to come to be known as "Doob Class" at some point, but it has stayed Class D for 40 years now, so we see no point in changing it. (There are no Classes A, B, and C.) We now come to our first version of the Doob-Meyer decomposition theorem. It is this theorem that gives the fundamental basis for the theory of stochastic integration. Theorem 7 (Doob-Meyer Decomposition: Case of Totally Inacces- sible Jumps). Let Z be a cadlag supermartingale with Zo = 0 of Class D, and such that all jumps of Z occur at totally inaccessible stopping times. Then there exists a unique, increasing, continuous, adapted process A with Ao = 0 such that Mt = Zt + At is a uniformly integrable martingale. We first give the proof of uniqueness which is easy. For existence, we will first establish three lemmas. Proof of uniqueness. Let Z = M - A and Z = N - C be two decompositions of Z. Subtraction yields M - N = A - C, which implies that M - N is a continuous martingale with paths of finite variation. We know however by the Corollary of Theorem 27 of Chap. II that M - N is then a constant martingale which implies Mt - Nt = M o - No = 0 - 0 = 0 for all t. Thus M = Nand A=C. 0 Lemma 1. Let IF be a discrete time filtration and let C be a non-decreasing process with Co = 0, and Ck E Fk-l. Suppose there exists a constant N > 0 such that E{Coo - CklFd ::; N a.s. for all k. Then E{C~} ::; 2N2 • Proof of Lemma 1. First observe E{Coo } = E{E{Coo - CoIFo}} ::; N. Let- ting Ck = Ck+! - Ck ~ 0, we obtain by rearranging terms: Cc:, = 22:)Coo - Ck)Ck - L c%. k;::::O k;::::O Thus it follows that 3 The Doob-Meyer Decompositions 107 E{C;',}:::; 2E(EE{C= - CklFdcd:::; 2NE(Ecd:::; 2NE{C=} k20 k20 Choose and fix a constant v E Z+ and let Dn = {k2-n : 0 :::; k2-n :::; v}. Lemma 2. Let T be a totally inaccessible stopping time. For 6 > 0, let R(6) = SUPt:'Ov P(t :::; T :::; t + 61Ft ). Then R(6) ----> 0 in probability as 6 ----> O. Proof of Lemma 2. Let a > 0 and Sn(6) = inft{t E Dn : P(t :::; T :::; t + 61Ft ) > a}!\v. First we assume Sn(6) is less than T. Since Sn(6) is countably valued, it is accessible, and since T is totally inaccessible, P(Sn(6) = T) = O. Suppose r c {T < t}, and also r E Ft. Then E{E{I{t:'OT:'OtH}IFt}Ir} = E{I{t:'OT:'OtH}Ir} = O. Suppose now P(T < Sn(6)) > O. Then for some t E Dn, P(T < t, Sn(6) = t) > O. Let r = {T < t, Sn(6) = t}. Then from the definition of Sn(6) we obtain E{I(t:'OT:'OtH)IFt}lr > aIr, (*) a contradiction. Thus we conclude Sn(6) < T a.s. Next we define a stopping time S as follows. Let 8(6) = infn Sn(6) and S = SUPn 8(~). Fix n, so that 8(~) = infk Sk(~)' Thus on {S = T}, 8(~) < S. Hence since S = sUPn 8(~), we have S is accessible on {S = T}, which implies that T is accessible on {S = T}, which in turn implies P(S = T) = O. Consequently E{I{s=T}IFs-} = 0 a.s. Suppose now that the result is not true. That is suppose there exists an c > 0 such that P(R(6) > a) > c, for all 6 tending to O. Let (3 > 0 and 6 < (3. For n sufficiently large P(E{I{Sn(8):'OT:'OSn(8)H}IFSn(8)} > a) ~ c. We also have P(T = 8(6)+6) = 0, since if not 8(6)+6-~ could announce part of T which would contradict that T is totally inaccessible. Thus P({Sn (6) :::; T:::; Sn(6) +6} ~ {8(6) :::; T:::; 8(6) +6}) ----> 0 as n ----> 00 where the symbol ~ denotes the symmetric difference set operation. Since the symmetric difference tends to 0, if P(Sn(6) :::; T :::; Sn(6) + 6IFSn(8) ~ a) ~ c for any 6 > 0, then we must also have since otherwise were it to tend to 0 along with the symmetric difference tend- ing to 0, then we would have (*) ----> 0, a contradiction. Thus we have P(8(6) :::; T:::; 8(6) + (3IFs(8)) > a on a set A, with P(A) ~ c. From the definition of S and working on the set A, this implies P(E{I{s:'OT:'Os+,6}IFs-} ~ a) ~ c. Let (3 ----> 0 to obtain P(T = SIFs-) ~ a, and We have a contradiction. 0 108 III Semimartingales and Decomposable Processes Lemma 3. Let the hypotheses of Theorem 7 hold and also assume IZI ::;; N where N > 0 is a constant, and further that the paths of Z are constant after a stopping time v. Let W(J) = SUPt a:~Zt > b}, Ii+l = inf{t > Ii : ~Zt > b}. Since IZI ::;; N be hypothesis, I~ZTJ ::;; 2N. Choose k such that P(Tk ::;; v) < E. Then k P(W+(J) > a) ::;; P(Tk ::;; v) + LP(supE{~ZTil{t:'OTi:'Ot+8}IFd > ~) i=l t k ::;; E+ LP(supP(t ::;;Ti ::;; t+JIFt ) > 2;N). i=l t By the previous lemma we know that R(J) -. 0 in L 2 , so by taking J small enough, we get the above expression less than 2E. The reasoning for W- is analogous. We achieve W(J) ::;; Wb(J)+ W+(J) +W-(J), which gives W(J) -. a in L 2 . 0 We return to the proof of Theorem 7. Proof of existence. First suppose the jumps of Z are bounded by a constant c. Let TN = inf{t > 0 : IZtl ~ N - c} 1\ N, and zf: = Zt/\TN" Then Izf:1 ::;; 3 The Doob-Meyer Decompositions 109 Iztl + c ::;: N, and Z N is constant after TN. Thus we are now reduced to the case considered by the Lemma 2. Fix n and let F kn = F k • Define2"" All the ak ~ a since Z is a supermartingale, and also ak E :FJ:-l. Let Ai: = 2:;=1 aj. Then Lk = Z 2"n + Ak is an :FJ: discrete time martingale. Define n An f k-l kB t = k i -- < t 110 III Semimartingales and Decomposable Processes loss of generality we assume Z has at most one jump greater than or equal to 2N in absolute value, that it occurs at the time T, and that Z is constant after the time T. We let z: = I: ~Zsl{C~zs2:2N}= ~ZTl{~zT2:2N}1{t2:T} s-:;,t Zt- = L~Zsl{~Zs-:;'-2N} = ~ZTl{~zT-:;'-2N.}1{t2:T} s-:;,t Since -Z+ and Z- both have decreasing paths, they are both supermartin- gales. Suppose we can show the theorem is true for -Z+ and Z-. Let -Z+ = M+ - A+ and Z- = M- - A- be the two decompositions, with A+ and A- both continuous. Then Z = Z +( -Z+ +A+) - (Z- +A-) = Z +M+ -M- is a supermartingale with jumps bounded by 2N. Let Z = £1 - A be the unique decomposition of Z which we now know exists. Then Z = Z + M+ - M- , and therefore Z = (£1 + M+ - M-) - A is the desired (unique) Doob-Meyer decomposition. Thus it remains to show that - Z+ and Z- both have Doob-Meyer de- compositions. First observe that I~ZTI ::; IZT-I + IZTI ::; N + IZTI E L 1 , and hence E{I~ZTI} < 00. Choose a, t: > O. Next choose R > N so large that E{I~ZTll{I6.ZTI2:R}}::; Ea. The cases for when the jump is positive and when the jump is negative being exactly the same except for a minus sign, we only treat the case where the jump is negative. Let zf = ~ZTl{t2:T}l{-6.zT>R}, and zt = Zt- - Zf. We define Bn-, BnR, B nd analogously to the way we defined B n in equation (*) above. We first show Br- converges uniformly in t in probability. We have: P(supIB~--B;"-I>a) (**) t ::; P(sup IB~d - B;"dl > ~) + P(sup IB~RI > ~3) + P(sup IB;"RI > ~) t 3 t t 3 The second term on the right side of (**) above is less than 3t:: The third term on the right side of (**) is shown to be less than 3t: similarly. Since IZ R I is bounded by R, the first term on the right side of (**) can be made arbitrarily small by taking m and n large enough, analogous to what we did at the beginning of the proof for B n . Thus as we did before in this proof, we can conclude that B~+ converges uniformly in t in probability as n ----> 00, and we denote its limit process by At. 3 The Doob-Meyer Decompositions 111 We prove continuity of A exactly as before. Finally by taking a subsequence nj such that Bnj+ converges almost surely, we get E{A;t,} = E{ lim B~+}::::; liminfE{B~+} = -E{Z~,} nj---+oo nj---+oo by Fatou's Lemma. From this it easily follows that Z+ + A+ is a uniformly integrable martingale, and the proof is complete. 0 While Theorem 7 is sufficient for most applications, the restriction to su- permartingales having jumps only at totally inaccessible stopping times can be insufficient for some needs. When we move to the general case we no longer have that A is continuous in general (however see Exercises 24 and 26 for supplemental hypotheses that assure that the increasing process A of the Doob-Meyer decomposition is in fact continuous). Without the continuity we lose the uniqueness of the decomposition, since there exist many martingales of finite variation (for example, the compensated Poisson process) that we can add to the martingale term and subtract from the finite variation term of a given decomposition, to obtain a second, new decomposition. Instead of the continuity of A we add the condition that A be predictably measurable. Theorem 8 (Doob-Meyer Decomposition: General Case). Let Z be a cadlag supermartingale with Zo = 0 of Class D. Then there exists a unique, increasing, predictable process A with Ao = 0 such that Mt = Zt + At is a uniformly integrable martingale. Before proving Theorem 8 let us introduce the concept of a natural process. We introduce two definitions and prove the important properties of natural processes in the next three theorems. Definition. An FV process A with Ao = 0 is of integrable variation if the expected total variation is finite: E{Ir~) IdAs I} < 00. A shorthand notation for this is E{IAloo} < 00. An FV process A is of locally integrable variation if there exists a sequence of stopping times (Tn )n21 increasing to 00 a.s. such Tn that E{Jo IdAsl} = E{IAITn } < 00, for each n. Definition. Let A be an (adapted) FV process, Ao = 0, of integrable varia- tion. Then A is a natural process if E{[M, A]oo} = 0 for all bounded martingales M. Here is the key theorem about natural processes. This use of natural pro- cesses was Meyer's original insight that allowed him to prove Doob's conjec- ture, which became the Doob-Meyer decomposition theorem. Theorem 9. Let A be an FV process, Ao = 0, and E{IAloo} < 00. Then A is natural if and only if 112 III Semimartingales and Decomposable Processes for any bounded martingale M. Proof. By integration by parts we have Then MoAo = 0 and letting Nt = J; As-dMs, we know that N is a local martingale (Theorem 20 of Chap. II). However using integration by parts we see that E{N~} < 00, hence N is a true martingale (Theorem 51 of Chap. I). Therefore E {Jooo A s- dMs } = E{ N oo} - E{ No} = 0, since N is a martingale, so that the equality holds if and only if E{[M, AJoo} = O. 0 Theorem 10. Let A be an FV process of integrable variation which is natural. If A is a martingale then A is identically zero. Proof. Let T be a finite stopping time and let H be any bounded, nonnegative martingale. Then E{JoT Hs_dAs} = 0, as is easily seen by approximating sums and the Dominated Convergence Theorem, since J: IdAsI E Ll and E{AT } = O. Using the naturality of A, E{HTAT } = E{foT Hs_dAs} = 0, and letting H t = E{1{Ar>0}IFd then shows that P(AT > 0) = o. Since E{AT } = 0, we conclude AT == 0 a.s., hence A == O. 0 Theorem 11. Let A be an FV process of integrable variation with Ao = o. If A is predictable, then A is natural. The proof of this theorem is quite intuitive and natural provided we accept a result from Chap. IV. (We do not need this theorem to prove the theorem we are using from Chap. IV.) Proof. Let M be a bounded martingale. First assume A is bounded. Then the stochastic integral Ir;" AsdMs exists, and it is a martingale by Theo- rem 29 in Chap. IV combined with, for example, Corollary 3 of Theorem 27 of Chap. II. Therefore E{fooo AsdMs} = E{AoMo} = 0, since A o = o. How- ever E{f;' As_dMs} = E{Ao_Mo} = 0 as well, since A o- = o. Further, 100 AsdMs - 100 As_dMs = 100 (As - A s- )dMs =100 LlAsdMs = 2: LlAsLlMs O 3 The Doob-Meyer Decompositions 113 since A is a quadratic pure jump semimartingale. Therefore Since M was an arbitrary bounded martingale, A is natural by definition. Finally we remove the assumption that A is bounded. Let An = n/\ (A V(-n)). Then An is bounded and still predictable, hence it is natural. For a bounded martingale M E{[M,A](X)} = limE{[M, An](X)} = 0, n by the Dominated Convergence Theorem. Therefore A is natural. o The next theorem is now obvious and should be called a Corollary at best. Because of its importance, however, we give it the status of a theorem. Theorem 12. Let M be a local martingale with paths of finite variation on compact time sets. If M is predictably measurable, then M is constant. That is, Mt = Mo for all t, almost surely. Proof. This theorem is a combination of Theorems 10 and 11. o Proof of Theorem 8 (Doob-Meyer: General Case). We begin with the unique- ness. Suppose Z = M - A and Z = N - C are two decompositions. By sub- traction we have M - N = A - C is a martingale with paths of finite variation which is predictable. By Theorem 12 we have that M - N is constant and since Mo - No = 0, it is identically zero. This gives the uniqueness. The existence of the decomposition is harder. We begin by defining stop- ping times Tn,j, where Tn,j is the j-th time I~Ztl is in the half-open interval bounded by 2-n and 2-(n-l), where -00 < n < 00. We decompose Tn,j into its accessible and totally inaccessible parts. Since we can cover the ac- cessible part with a sequence of predictable times, we can thus separate each Tn,j into a totally inaccessible time and a sequence of predictable times, with disjoint (stopping time) graphs. Therefore, by renumbering, we can obtain a sequence of stopping times (Sik::l with disjoint graphs, each one of which is either predictable or totally inaccessible, which exhaust the jumps of Z (in the sense that the jump times of Z are contained in the union of the graphs of the Si) and are such that for each i there exists a constant bi such that bi < I~ZSil :::; 2bi . We define Zo(t) = Zt and inductively define Zi+l(t) = Zi(t) and Ai(t) = o if Si is totally inaccessible; whereas Ai(t) = -E{~ZsJFsi- }1{si:O;t} and then Zi+l(t) = Zi(t) + Ai(t) in the case where Si is predictable (note that ~ZSi ELI). We will show that: each Ai is increasing; Zi is a supermartingale for each i; and E{L:j=l Aj(oo)} :::; C, for each i, where C is a constant not depending on i. We will prove these three properties by induction. We begin by showing that each Ai is increasing. This is trivial if Si is totally inaccessible. Let then 114 III Semimartingales and Decomposable Processes Si be a predictable time. Let Sf be an announcing sequence for Si. Since each Zi is a supermartingale, and using the Martingale Convergence Theorem, we have E{~Zi(Si)IFsi-}= limnE{~Zi(Si)IFsi}. Now fix n: E{~Zi(Si)IFsn}= limE{Zi(Si) - Zi(Sf)IFs:,} , k ' = limE{E{Zi(Si) - Zi(Sf)IFsdIFs :,}k ' , :::; 0, and thus Ai is increasing. To see that Zi+1 is a supermartingale, it will suffice to show that whenever U1 and U2 are stopping times with U1 :::; U2 then E{Zi+1(UI)} :::=: E{ZiH(U2)}. Again letting Sf be an announcing sequence for Si we have E{Zi(UI)} - E{Zi(U2)} = E{Zi(U1)} - E{Zi((Ul V Sf) 1\ U2)} + E{Zi((Ul V Sf) 1\ U2)} - E{Zi((Ul V Si) 1\ U2)} + E{Zi((Ul V Si) 1\ U2)} - E{Zi(U2)} where each of the summands on the right side above are nonnegative. Then let n ----> 00 to get E{Zi(U1)} - E{Zi(U2)}:::=: O. Finally we want to show that E{L~=lAj(oo)} :::; C. It is easy to check (see Exercise 1 for example) that the minimum of predictable stopping times is predictable, hence we can order the predictable times so that Sl < S2 < ... on the set where they are all finite. Let Ai be the collection of all j such that j :::; i and such that the Sj are all predictable. Then -E{"" ~Zs} = lim"" E{ZSk - Zs}~ J k~ J J Ai :::; lifL(E{Zs;vSj_J + E{ZSj_l - ZS;VSj_J Ai +E{ZSi - Zoo} + E{Zo - Zs:}) = E{Zo - Zoo}, which is bounded by a constant not depending on i. To complete the proof, we note that because the processes Ai are in- creasing in t and the expectation of their sum is bounded independently of i, we have that L~l Ai(t) converges uniformly in t a.s., as h ----> 00, and we call the limit Aoo(t). Fatou's Lemma gives us that Aoo(t) is integrable. Each A(t) is easily seen to be predictable, and hence Aoo is the almost sure limit of predictable measurable processes, so it too is predictable. Next set Zoo(t) = Zt + Aoo(t) = limi Zi(t). Since each Zi is a supermartingale, cou- pled with the uniform convergence in t of the partial sums of the Ai pro- cesses, we get that Zoo(t) is a supermartingale. Moreover since each Zi is dtdlag, again using the uniform convergence of the partial sums we obtain 3 The Doob-Meyer Decompositions 115 that Zoo is also cadlag. Since the partial sums are uniformly bounded in ex- pectation, Zoo(t) is of Class D. Finally by our construction of the stopping times Si, E{~Zoo(T)IFT-} = 0 for all predictable times T. We can then show as before that W(J) ----> a in probability as J ----> a for Zoo, and we ob- tain Zoo(t) = Mt - A(t). The process At = Aoo(t) + A(t) is then the desired increasing predictable process. 0 The next theorem can also be considered a Doob-Meyer decomposition theorem. It exchanges the uniform integrability for a weakening of the con- clusion that M be a martingale to that of M being a local martingale. Theorem 13 (Doob-Meyer Decomposition: Case Without Class D). Let Z be a cadlag supermartingale. Then Z has a decomposition Z = Zo + M - A where M is a local martingale and A is an increasing process which is predictable, and M o = Ao = O. Such a decomposition is unique. Moreover if limt--->oo E{Zt} > -00, then E{Aoo } < 00. Proof. First consider uniqueness. Let Z = Zo + M - A and Z = Zo + N - C be two decompositions. Then M - N = A - C by subtraction. Hence A - C is a local martingale. Let M Tn be a uniformly integrable martingale. Then and therefore E{Ar} :'S: E{Zo - Zt}, using Theorem 17 of Chap. 1. Letting n tend to 00 yields E{At} :'S: E{Zo - Zt}. Thus A is integrable on [0, to], each to, as is C. Therefore A - C is of locally integrable variation, predictable, and a local martingale. Since Ao - Co = 0, by localization and Theorems 10 and 11, A - C = O. That is, A = C, and hence M = N as well and we have uniqueness. Next we turn to existence. Let Tm = inf{t : IZtl ;:::: m} 1\ m. Then T m in- crease to 00 a.s. and since they are bounded stopping times ZTTn ELI each m (Theorem 17 of Chap. I). Moreover the stopped process ZTTn is dominated by the integrable random variable max(IZTTnI, m). Hence if X = ZTTn for fixed m, then 1t = {XTj T a stopping time} is uniformly integrable. Let us implicitly stop Z at the stopping time Tm and for n > a define y;;n = Zt - E{ZnIFtl, with y;;n = Y;;';,n = 0 when t ;:::: n. Then yn is a positive supermartingale of Class D and hence y;;n = Yon + Mtn - Af. Letting N!: = E{ZnIFt}, a martingale, we have on [0, n] that To conclude, therefore, it suffices to show that Ar' = Af on [0, n], for m ;:::: n. This is a consequence of the uniqueness already established. The uniqueness also allows us to remove the assumption that Z is stopped at the time T m . Finally, note that since E{At} :'S: E{Zo - Zt}, and since A is increasing, we have by the Monotone Convergence Theorem 116 III Semimartingales and Decomposable Processes E{Aoo } = lim E{Atl::::; lim E{Zo - Ztlt-+oo t-+oo which is finite if limt-+oo E{Ztl > -00. 4 Quasimartingales o Let X be a cadlag, adapted process defined on [0,00].1 Definition. A finite tuple of points T = (to, tl, ... , tn+l) such that °= to < tl < ... < tn+l = 00 is a partition of [0,00]. Definition. Suppose that T is a partition of [0, 00] and that X ti E L l , each ti E T. Define n C(X, T) = 2: IE{Xti - Xt;+lIFtJI· i=O The variation of X along T is defined to be The variation of X is defined to be Var(X) = sup Varr(X), r where the supremum is taken over all such partitions. Definition. An adapted, cadHtg process X is a quasimartingale on [0,00] if E{IXt !} < 00, for each t, and if Var(X) < 00. Before stating the next theorem we recall the following notational conven- tion. If X is a random variable, then X+ = max(X, 0), X- = -min(X, 0). Also recall that by convention if X is defined only on [0, (0), we set X oo = O. Theorem 14. Let X be a process indexed by [0, (0). Then X is a quasi- martingale if and only if X has a decomposition X = Y - Z where Y and Z are each positive right continuous supermartingales. 1 It is convenient when discussing quasimartingales to include 00 in the index set, thus making it homeomorphic to [0, t] for °< t ~ 00. If a process X is defined only on [0,00) we extend it to [0,00] by setting X oo = o. 4 Quasimartingales 117 Proof For given 8 ~ 0, let 2:(8) denote the set of finite subdivisions of [8,00]. For each T E 2:(8), set where C(X,T)+ denotes 2:tiETE{Xti - Xti+JFtJ+, and analogously for C (X, T) -. Also let --< denote the ordering of set containment. Suppose (J, T E 2:(8) with (J --< T. We claim YsCJ" :::; YST a.s. To see this let (J = (to, ... , tn). It suffices to consider what happens upon adding a subdivision point t before to, after tn, or between t i and ti+1. The first two situations being clear, let us consider the third. Set A = E{Xti - XtlFdi B = E{Xt - Xti+1IFtl; C = E{Xti - Xti+1IFt;}i then C = A + E{BIFd, hence C+ :::; A+ + E{BIFd+ :::; A+ + E{B+IFtj, by Jensen's inequality. Therefore and we conclude YsCJ" :::; YST. Since E{YST} is bounded by Var(X), taking limits in £1 along the directed ordered set 2:(8) we define Ys = lim YST , T and we can define Zs analogously. Taking a subdivision with to = 8 and tn +! = 00, we see YST - Z; = E{C+ - C-IFs } = X s , and we deduce that Ys - Zs = X S' Moreover if 8 < t it is easily checked that Ys ~ E{YtIFs } and Zs ~ E{ZtIFs}. Define the right continuous processes Yf == Yt+, Zt == Zt+, with the right limits taken through the rationals. Then Y and Z are positive supermartingales and Ys - Zs = X S ' For the converse, suppose X = Y - Z, where Y and Z are each positive supermartingales. Then for a partition T of [0, t] :::; E{L E{Yfi - Yfi+1IFti }} + E{L E{ Zti - Zti+lIFti}} tiET tiET = E{Yo} + E{Zo} - (E{Yf} + E{Zd)· Thus X is a quasimartingale on [0, t], each t > 0. D 118 III Semimartingales and Decomposable Processes Theorem 15 (Rao's Theorem). A quasimartingale X has a unique decom- position X = M + A, where M is a local martingale and A is a predictable process with paths of locally integmble variation and Ao = O. Proof This theorem is a combination of Theorems 13 and 14. 5 Compensators o Let A be a process of locally integrable variation, hence a fortiori an FV process. A is then locally a quasimartingale, and hence by Rao's Theorem (Theorem 15), there exists a unique decomposition where A is a predictable FV process. In other words, there exists a unique, predictable FV process A such that A - A is a local martingale. Definition. Let A be an FV process with Ao = 0, _with locally inteiQ"able total variation. The unique FV predictable process A such that A - A is a local martingale is called the compensator of A. The most common examples are when A is an increasing process locally of integrable variation. When A is an increasing process it is of course a submartingale, and thus by the Doob-Meyer Theorem we know that its com- pensator A is also increasing. We also make the obvious observation that E{Atl = E{Atl for all t, 0:::; t :::; 00. Theorem 16. Let A be an increasing process of integmble variation, and let HElL be such that E{f; HsdAs} < 00. Then, Proof Since A - A is a martingale, so also is f Hsd(As - As), and hence it has constant expectation equal to o. 0 In Chap. IV we develop stochastic integration for integrands which are predictable, and Theorem 16 extends with HElL replaced with H predictable. One of the simplest examples to consider as an illustration is that of the Poisson process N = (Ntk~_o with parameter A. Recall that Nt - At is a martingale. Since the process At = At is continuous and obviously adapted, it is predictable (natural). Therefore Nt = At, t ~ o. A natural extension of the Poisson process case is that of counting processes without explosions. We begin however with a counting process that has only one jump. Let TJ be the counting process TJt = l{t~T}' where T is a nonnegative random variable. Let JF be the minimal filtration making T a stopping time. 5 Compensators 119 Theorem 17. Let peT = 0) = °and peT > t) > 0, each t > 0. Then the JF compensator A of"7, where "7t = 1{t2T }' is given by rAt 1 At = Jo I_F(u_)dF(u), where F is the cumulative distribution function of T. If T has a diffuse distribution (that is, if F is continuous), then A is continuous and At = -In(l - F(T 1\ t)). Before proving Theorem 17 we formalize an elementary result as a lemma. Lemma. Let (n, F, P) be a complete probability space. In addition, suppose T is a positive F measurable random variable and 110 = a{T 1\ t}, where 11 = 110 V N, N are the null sets of F, and F t = nu >t:F2. Then JF so constructed is the smallest filtration making T a stopping time. Let Y E L1(F). Then Proof. By the hypotheses on T the a-algebra 11 is equal to the Borel a- algebra on [0, t] together with the indivisible atom (t, 00). Observe that 11 = a{T 1\ U; u ~ t}, and the result follows easily. D Proof of Theorem 17. Fix to > °and let 1rn be a sequence of partitions of [0, to] with lillln--->oo mesh(1rn ) = 0. Define Ar = L:1rn E{"7ti+l - "7ti IFtJ for °~ t ~ to. Then E{ IF.} E{l{T>t;} "7ti+,} "7ti+l ti = "7ti+l 1{T::;ti} + peT > ti) l{T>ti} by the lemma preceding this proof. The first term on the right above is "7ti+l1{ti2T} = 1{t;+12T }1{ti2T} = 1{ti2T}' Furthermore, l{T>ti}"7ti+l = 1{T>t;}1{ti+12T} = l{ti 120 III Semimartingales and Decomposable Processes which are Riemann sums converging to f~1\T(1 - F(u- ))-ldF(u). The re- maining claims of the theorem are obvious. 0 Corollary 1. If F in Theorem 17 is absolutely continuous, then so also is A. Corollary 2. If T in Theorem 17 has an exponential distribution, then At = T 1\ t. Proof. Since in this case F is continuous and a distribution function, we can represent it as F(x) = 1 - e-q,(x). We have that At = -In(1 - F(T 1\ t)) and equivalently we can write At = ¢(T 1\ t). In the case of this corollary, the function ¢(x) = x, and the result follows. 0 Remark. Theorem 17 gives an explicit formula for the compensator of a process'TJt = 1{t2 T } when one uses the minimal filtration making T a stopping time. A perhaps more interesting question is the following: suppose that one has a non-trivial filtration IF satisfying the usual hypotheses, and a strictly positive random variable L that is not a stopping time. Can one expand the filtration to make a larger filtration G that renders LaG stopping time, and then calculate the G compensator of fJt = l{t2 L }? This has a positive answer with the right hypotheses. To see how to do this, let X t = l[O,L)(t), which is not an IF adapted process. Let Z denote the optional projection of X onto IF. We prove in Chap. VI that Z is a supermartingale. Let Z = M -A be its Doob- Meyer decomposition with A predictable. (Note that M t = E{AooIFt }.) Using the techniques presented in Chap. VI, one can show that if L is "the end of an IF predictable set," then the G compensator of fJt = l{t2 L } is At, which is an IF predictable process! By the end of a predictable set, we mean that there exists an IF predictable set A c [0, ooJ x n such that L(w) = sup{t :::; 00 : (t,w) E A}. In Chap. VI we discuss times that are the ends of optional sets, known as "honest times," and this is of course less restrictive than being the end of a predictable set, since any predictable set is also optional.2 See [115] for this result described here and related ones. Theorem 17 treats the case of a point process with one jump. A special case of interest is counting processes. We have the following. Theorem 18. Let N be a counting process without explosions, adapted to a filtration G satisfying the usual hypotheses. Then the compensator of N, call it A, always exists. Proof. Since N has non-decreasing paths, to ensure the existence of a com- pensator we need only to show that N is locally of integrable variation. Let Tn be the time of the n-th jump of N. Since N has no explosions, the times 2 See Exercise 4. 5 Compensators 121 Tn increase to 00 a.s. Moreover INfI',Tn I= Ntl\Tn ::::: n and thus N is locally of bounded variation, hence certainly locally of integrable variation. 0 Let Nt = Ei>l 1{t2T;} be a counting process without explosions and let Si = Ti - Ti- l be its interarrival times, where we take To = 0. Let :Pi = a{Ns ; S ::::: t}, and let Ft = :Pi V N, the filtration completed by the F-null sets, as usual. (Note that this filtration is already right continuous.) Let us define the cumulative distribution functions for the interarrival times slightly informally as follows (note that we are using the unusual format of peS > x) rather than the customary P(S::::: x)): F1(t) = P(SI > t) Pi(Sl, ... , Si-l; t) = P(Si > tlSI = Sl,··., Si-l = Si-l) where the Sj are in [0,00], j ~ 1. (If one of the Sj takes the value +00, then the corresponding cumulative distribution function is concentrated at +00.) Define i t -1ePi(Sl"",Si-l;t)= F( . )dFi(SI"",Si-l;S).o iSl,···,Si-l,S- We now have the tools to describe the compensator of N for its minimal completed filtration. We omit the proof, since it is simply a more notationally cumbersome version of the proof of Theorem 17. Theorem 19. Let N be a counting process without explosions and let IF be its minimal completed filtration. Then the compensator A of N is given by At = ePI(SI) + eP2(SI; S2) + ... + ePi-I(SI, ... , Si-2; Si-l) + ePi(Sl, ... ,Si-l; t - Ti ) on the event {Ti < t ::::: Ti+d. Corollary. Let N be a counting process without explosions and independent interarrival times. Then the functions ePi defined in equation (*) have the simplified form and the compensator A is given by i-I At = 2::[2:: ePj(Sj) + ePi(t - Ti)]l{T;~t 122 III Semimartingales and Decomposable Processes variable with a continuous density for its distribution function P, given by f. The hazard rate>. is defined, when it exists, to be 1 >'(t) = lim -hP(t :::; T < t + hiT 2 t). h-+o The intuition is that this is the probability the event will happen in the next infinitesimal unit of time, given that it has not yet happened. Another way of viewing this is that P(t :::; T < t + hiT 2 t) = >'h + o(h). By Theorem 17 we have that the compensator of Nt = l{t2T} is At = f~I\T >.(s)ds. Or in language closer to stochastic integration, tl\T Nt - io >.(s)ds is a local martingale. In medical trials and other applications, one is faced with situations where one is studying random arrivals, but some of the data cannot be seen (for example when patients disappear unexpectedly from clinical trials). This is known as arrivals with censored data. A simple example is as follows. Let T be a random time with a density and let U be another time, with an arbitrary distribution. T is considered to be the arrival time and U is the censoring time. Let X = T 1\ U, Nt = l{t2 x }l{u2T}, and NF = l{t2 x }I{T>u} with Ft = a{Nu,N;;j u:::; t}. Let>' be the hazard rate function for T. >. is known as the net hazard rate. We define >.#, known as the crude hazard rate, by >.# = ~~ ~P(t:::; T < t + hiT 2 t, U 2 t) when the limit exists. We then have that the compensator of N is given by At = f~ l{x2 u }>,#(u)du = f~l\x >.#(u)du = f~I\TI\U >.#(u)du, or equivalently Nt -It l{x2u }>,#(u)du is a local martingale. If we impose the condition that>' = >.#, which can be intuitively interpreted as P(s :::; T < s + dslT 2 s) = P(s :::; T < s + dslT 2 s, U 2 s), then we have the satisfying result that Nt -It I{X2 u }>'(u)du is a local martingale. As we saw in Chap. II, processes of fundamental importance to the the- ory of stochastic integration are the quadratic variation processes [X, X] = ([X, X]t)t20, where X is a semimartingale. Definition. Let X be a semimartingale such that its quadratic variation process [X, X] is locally integrable. Then the conditional quadratic vari- ation of X, denoted (X,X) = ((X,X)t)t>o, exists and it is defined to be the -~ compensator of [X, X]. That is (X, X) = [X,X]. 5 Compensators 123 If X is a continuous semimartingale then [X, X] is also continuous and hence already predictable; thus [X, X] = (X, X) when X is continuous. In particular for a standard Brownian motion B, [B, B]t = (B, B)t = t, all t ~ O. The conditional quadratic variation is also known in the literature by its notation. It is sometimes called the sharp bracket, the angle bracket, or the oblique bracket. It has properties analogous to that of the quadratic variation processes. For example, if X and Yare two semimartingales such that (X, X), (Y, Y), and (X + Y, X + Y) all exist, then (X, Y) exists and can be defined by polarization 1(X, Y) = -((X + Y,X + Y) - (X,X) - (Y, Y)). 2 However (X, Y) can be defined independently as the compensator of [X, Y] provided of course that [X, Y] is locally of integrable variation. In other words, there exist stopping times (Tn)n>l increasing to 00 a.s. such that E{foTn Id[X, Y]sl} < 00 for each n. Als~, (X, X) is a non-decreasing pro- cess by the preceding discussion, since [X, X] is non-decreasing. The condi- tional quadratic variation is inconvenient since unlike the quadratic variation it doesn't always exist. Moreover while [X, X], [X, Y], and [Y, Y] all remain invariant with a change to an equivalent probability measure, the sharp brack- ets in general change with a change to an equivalent probability measure and may even no longer exist. Although the angle bracket is ubiquitous in the literature it is sometimes unnecessary as one can often use the quadratic vari- ation instead, and indeed whenever possible we use the quadratic variation rather than the conditional quadratic variation (X, X) of a semimartingale X in this book. Nevertheless the process (X, X) occurs naturally in extensions of Girsanov's theorem for example, and it has become indispensable in many areas of advanced analysis in the theory of stochastic processes. We end this section with several useful observations that we formalize as theorems and a corollary. Note that the second theorem below is a refinement of the first (and see also Exercise 25). Theorem 20. Let A be an increasing process of locally integrable variation, all of whose jumps oCCur at totally inaccessible stopping times. Then its com- pensator A is continuous. Theorem 21. Let A be an increasing process of locally integrable variation, and let T be a jump time of A which is totally inaccessible. Then its compen- sator A is continuous at T. Proof. Both theorems are simple consequences of Theorem 7. o Corollary. Let A be an increasing predictable process of locally integrable variation and let T be a stopping time. If P(AT #- AT -) > 0 then T is not totally inaccessible. 124 III Semimartingales and Decomposable Processes Proof Suppose T were totally inaccessible. Let A be the compensator of A. Then A is continuous at T. But since A is already predictable, A = A, and we have a contradiction by Theorem 21. 0 Theorem 22. Let T be a totally inaccessible stopping time. There exists a martingale M with paths of finite variation and with exactly one jump, of size one, occurring at time T (that is, M T i=- M T- on {T < oo}). Proof Define Ut = l{t~T}' Then U is an increasing, bounded process of integrable variation, and we let A = if be the compensator of U. A is continuous by Theorem 20, and M = U - A is the required martingale. 0 6 The Fundamental Theorem of Local Martingales We begin with two preliminary results. Theorem 23 (Le Jan's Theorem). Let T be a stopping time and let H be an integrable random variable such that E{HIFT-} = 0 on {T < oo}. Then the right continuous martingale H t = E{HIFtl is zero on [O,T) = {(t,w) : 0::::; t < T(w)}. Proof Since the martingale (Hdt>o is right continuous it suffices to show that Htl{t 6 The Fundamental Theorem of Local Martingales 125 Theorem 25 (Fundamental Theorem of Local Martingales). Let M be a local martingale and let {3 > o. Then there exist local martingales N, D such that D is an FV process, the jumps of N are bounded by 2{3, and M=N+D. Proof. We first set Ct = L I~M811{I~MsI2,8}, 0 o. Let R = TA . By Theorem 3 we can decompose R as R = RA 1\ RB, where RA is accessible and RB is totally inaccessible. Then P(RB < 00) = 0 by Theorem 7, whence 126 III Semimartingales and Decomposable Processes R = RA. Let (Tk )k>l be a sequence of predictable times enveloping R. It will suffice to show ILiNTk I ::::: 2{3, for each Tk. Thus without loss of generality we can take R = T to be predictable. By convention, we set tlNT = °on {T = oo}. Since we know that tlAT E FT- by Theorem 24, we have tlNT = tlNT - E{tlNTIFT-} = tl(M - Ah + tlAT - E{tl(M - AhIFT-} - E{tlATIFT-} = tl(M - A)T - E{tl(M - A)TIFT-}. Since Itl(M - Ahl ::::: {3, the result follows. o If the jumps of a local martingale M with M o = 0 are bounded by a constant {3, then M itself is locally bounded. Let Tn = inf{t : IMtl ?: n}. Then IMti\Tn I ::::: n + {3. Therefore M is a fortiori locally square integrable. We thus have a corollary. Corollary. A local martingale is decomposable. Of course if all local martingales were locally square integrable, they would then be trivially decomposable. The next example shows that there are mar- tingales that are not locally square integrable (a more complex example is published in Doleans-Dade [50]). Example. Let (n, F, P) be complete probability space and let X be a random variable such that X E £l, but X 'I- £2. Define the filtration :Pi = {{0,n}, F, 0::::: t < 1, t ?: 1, where Ft =:Pi VN, with N all the P-null sets of F. Let Mt = E{XIFt}, the right continuous version. Then M is not a locally square integrable martingale. The next example shows another way in which local martingales differ from martingales. A local martingale need not remain a local martingale under a shrinkage of the filtration. They do, however, remain semimartingales and thus they still have an interpretation as a differential. Example. Let Y be a symmetric random variable with a continuous distri- bution and such that E{IYI} = 00. Let X t = Yl{t2 1}' and define g~ = {a{IYI}, a{Y}, 0::::: t < 1, t ?: 1, where G = (gt)t>o is the completed filtration. Define stopping times Tn by Tn = {o, 00, if WI ?: n, otherwise. 7 Classical Semimartingales 127 Then Tn reduce X and show that it is a local martingale. However X is not a local martingale relative to its completed minimal filtration. Note that X is still a semimartingale however. The full power of Theorem 25 will become apparent in Sect. 7. 7 Classical Semimartingales We have seen that a decomposable process is a semimartingale (Theorem 9 of Chap. II). We can now show that a classical semimartingale is indeed a semimartingale as well. Theorem 26. A classical semimartingale is a semimartingale. Proof. Let X be a classical semimartingale. Then X t = Mt + At where M is a local martingale and A is an FV process. The process A is a semimartin- gale by Theorem 7 of Chap. II, and M is decomposable by the corollary of Theorem 25, hence also a semimartingale (Theorem 9 of Chap. II). Since semi- martingales form a vector space (Theorem 1 of Chap. II) we conclude X is a semimartingale. 0 Corollary. A cadlag local martingale is a semimartingale. Proof. A local martingale is a classical semimartingale. Theorem 27. A cadlag quasimartingale is a semimartingale. o Proof. By Theorem 15 a quasimartingale is a classical semimartingale. Hence it is a semimartingale by Theorem 26. 0 Theorem 28. A cadlag supermartingale is a semimartingale. Proof. Since a local semimartingale is a semimartingale (corollary to The- orem 6 of Chap. II), it suffices to show that for a supermartingale X, the stopped process xt is a semimartingale. However for a partition T of [0, tJ, tiEr tiEr tiEr Therefore X t is a quasimartingale, hence a semimartingale by Theorem 27. 0 Corollary. A submartingale is a semimartingale. 128 III Semimartingales and Decomposable Processes We saw in Chap. II that if X is a locally square integrable local martingale and HElL, then the stochastic integral H . X is also a locally square inte- grable local martingale (Theorem 20 of Chap. II). Because of the corollary of Theorem 25 we can now improve this result. Theorem 29. Let M be a local martingale and let HElL. Then the stochastic integral H . M is again a local martingale. Proof A local martingale is a semimartingale by the corollary of Theorem 25 and Theorem 9 of Chap. II; thus H . M is defined. By the Fundamental Theorem of Local Martingales (Theorem 25) for (3 > 0 we can write M = N + A where N, A are local martingales, the jumps of N are bounded by (3, and A has paths of finite variation on compacts. Since N has bounded jumps, by stopping we can assume N is bounded. Define T by T = inf{t > 0 :I t IdAsl > m}. Then E{ft\T IdAsl} ~ m + (3 +E{I~MTI} < 00, and thus by stopping A can be assumed to be of integrable variation. Also by replacing H by H S 1{s>o} for an appropriate stopping time S we can assume without loss of generality that H is bounded, since H is left continuous. We also assume without loss that Mo - No = Ao = O. We know H· N is a local martingale by Theorem 20 of Chap. II, thus we need show only that H . A is a local martingale. Let an be a sequence of random partitions of [0, t] tending to the identity. Then L HTn (AT,n+ 1 - AT,n) tends to (H· A)t in ucp, where an is the sequence o = To ~ 1'1' ~ ... ~ Tr ~ .... Let (nk) be a subsequence such that the sums converge uniformly a.s. on [0, t]. Then by Lebesgue's Dominated Convergence Theorem. Since the last limit above equals (H· A)s, we conclude that H . A is indeed a local martingale. 0 A note of caution is in order here. Theorem 29 does not extend completely to processes that are not in IL but are only predictably measurable, as we will see in Emery's example of a stochastic integral behaving badly on page 176 in Chap. IV. Let X be a classical semimartingale, and let X t = X o + Mt + At be a decomposition where M o = Ao = 0, M is a local martingale, and A is an FV 7 Classical Semimartingales 129 process. Then if the space (n, F, (Ftk=:o, P) supports a Poisson process N, we can write as another decomposition of X. In other words, the decomposition of a classi- cal semimartingale need not be unique. This problem can often be solved by choosing a certain canonical decomposition which is unique. Definition. Let X be a semimartingale. If X has a decomposition X t = Xo + M t + At with M o = A o = 0, M a local martingale, A an FV process, and with A predictable, then X is said to be a special semimartingale. To simplify notation we henceforth assume Xo = 0. Theorem 30. If X is a special semimartingale, then its decomposition X = M + A with A predictable is unique. Proof. Let X = N +B be another such decomposition. Then M -N = B-A, hence B - A is an FV process which is a local martingale. Moreover, B - A is predictable, and hence constant by Theorem 12. Since Bo - Ao = 0, we conclude B = A. 0 Definition. If X is a special semimartingale, then the unique decomposition X = M + A with Mo = Xo and A o = °and A predictable is called the canonical decomposition. Theorem 15 shows that any quasimartingale is special. A useful sufficient condition for a semimartingale X to be special is that X be a classical semi- martingale, or equivalently decomposable, and also have bounded jumps. Theorem 31. Let X be a classical semimartingale with bounded jumps. Then X is a special semimartingale. Proof. Let X t = Xo + Mt + At be a decomposition of X with Mo = A o = 0, M a local martingale, and A an FV process. By Theorem 25 we can then also write X t = Xo + Nt + Bt where N is a local martingale with bounded jumps and B is an FV process. Since X and N each have bounded jumps, so also does B. Consequently, it is locally a quasimartingale and therefore decomposes where L is a local martingale and B is a predictable FV process (Theorem 15). Therefore X t = Xo + {Nt + Lt } + Bt is the canonical decomposition of X and hence X is special. o 130 III Semimartingales and Decomposable Processes Corollary. Let X be a classical semimartingale with continuous paths. Then X is special and in its canonical decomposition the local martingale M and the FV process A have continuous paths. Proof. X is continuous hence trivially has bounded jumps, so it is special by Theorem 31. Since X is continuous we must have flMT = -flAT for any stopping time T (flAT = 0 by convention on {T = oo}). Suppose A jumps at a stopping time T. By Theorem 3, T = TA 1\ TB, where TA is accessible and TB is totally inaccessible. By Theorem 21 it follows that P(lflATB I > 0) = o. Hence without loss of generality we can assume T is accessible. It then suffices to consider T predictable since countably many predictable times cover the stopping time T. Let Sn be a sequence of stopping times announcing T. Since A is predictable, we know by Theorem 24 that flAT is FT- measurable. Therefore flMT is also FT- measurable. Stop M so that it is a uniformly integrable martingale. Then flMT = E{flMTIFT_} =0, and M, and hence A, are continuous, using Theorem 7 of Chap. I. 0 Theorem 31 can be strengthened, as the next two theorems show. The criteria given in these theorems are quite useful. Theorem 32. Let X be a semimartingale. X is special if and only if the process Jt = sUPsst IflXsl is locally integrable. Theorem 33. Let X be a semimartingale. X is special if and only if the process X; = supsst IXsI is locally integrable. Before proving the theorems, we need a preliminary result, which is inter- esting in its own right. Theorem 34. Let M be a local martingale and let M; = sUPs 8 Girsanov's Theorem 131 Proof of Theorems 32 and 33. Since X is special the process A of its canonical decomposition X = M + A is of locally integrable variation. Writing Xt :$ Mt +1~ IdAs I, and since Mt is locally integrable by Theorem 34 and since A is of locally integrable variation, we have that X* is locally integrable. Further, note that I~Xt I = IXt - Xt-I :$ 21X;I, hence Jt :$ Xt and we have that Jt is also locally integrable. For the converse, it will suffice to show that there exists a decomposition X = M + A where A is of locally integrable variation, since then we can take the compensator of A and obtain a canonical decomposition. To this end note that I~As[ :$ I~Ms[ + I~Xsl :$ 2M; + Js, which is locally integrable by the hypotheses of Theorem 32 together with Theorem 34. Since Jt :$ Xt we have that I~As I is locally integrable by the hypotheses of Theorem 33 together with Theorem 34 as well. To complete the proof, let Tn = inf{t > 0 : 1~ IdAs I 2: n}, and note that 1{n [dAsl = 1oTn-[dAsl + I~ATnl :$ n + sUPs 132 III Semimartingales and Decomposable Processes If Q ~ P, then there exists a random variable Z in L 1 (dP) such that ~ = Z and Ep{Z} = 1, where E p denotes expectation with respect to the law P. We let Zt = E p { ~~ 1Ft} be the right continuous version. Then Z is a uniformly integrable martingale and hence a semimartingale (by the corollary of Theorem 26). Note that if Q is equivalent to P, then ~~ E L 1 (dQ) and ~~ = (~~) -1. We begin with a simple lemma, the easy proof of which we leave as an exercise for the reader. Lemma. Let Q rv P, and Zt = Ep {~IFt}. An adapted, cadlag process M is a Q local martingale if and only if M Z is a P local martingale. Proof. See Exercise 20. o Theorem 35 (Girsanov-Meyer Theorem). Let P and Q be equivalent. Let X be a classical semimartingale under P with decomposition X = M + A. Then X is also a classical semimartingale under Q and has a decomposition X = L + C, where Lt = Mt - it ~s d[Z, M]s is a Q local martingale, and C = X - L is a Q FV process. Proof. Recall that by Theorem 2 of Chap. II it is trivial that X is a Q semi- martingale. We need to show it is a classical semimartingale, with the above decomposition being valid. Since M and Z are P local martingales, they are semimartingales (corol- lary of Theorem 26) and is a local martingale as well (Theorem 29). Using integration by parts we have is also a P local martingale. Since Z is a version of Ep{~IFt},we have -2- is a cadlag version of EQ {~~ 1Ft }; therefore -2- is a Q semimartingale. Since P ~ Q, it is also a P semimartingale. Multiplying equation (*) by -2- we have M- (~) [Z,M] = ~ (/ Z_dM +f M_dZ). Note that if we were to multiply the right side of (**) by Z we would obtain a P local martingale. We conclude by the lemma preceding this theorem that 8 Girsanov's Theorem 133 M - (-2 )[Z, M] is a Q local martingale. We next use integration by parts (under Q): (~) [Z,M] = J ;_ d[Z,M] + J[Z,M]_d(~) +[[Z,M],~]. Let N = J[Z, M] _ d ( -2). Since -2 is a Q local martingale, so also is N (The- orem 29). The above becomes: ( 1 ) r t 1 1Zt [Z, M]t = Jo Zs- d[Z, M]s + Nt + [[Z, M], zlt = rZl_ d[Z,M]s +Nt + L ~ (~ ) ~[Z,M]sJo s O 134 III Semimartingales and Decomposable Processes where both N = J~ Zs_dXs + J~ Xs_dZs and M = N + [Z,XJ - (X, Z) are P local martingales. Therefore ZX - (X, Z) is a P local martingale. Let A = (X, Z), which of course is a predictable FV process. Moreover we know that A does not jump at totally inaccessible times, and that we can cover its jumps with predictable times. This implies that A is locally bounded. Let Tn = inf{t > 0 : IAtl > n}, and let ATn- be defined by ATn- = J l[o,Tn)(s)dAs. From now on we assume A = ATn-. Then A is still an FV process and is still predictable, IAI :$ n, and the stopping times Tn increase to 00 a.s. Integration by parts gives us also .If we show that [A, ~1is also a Q local martingale, we will be done. It suffices to show that EQ {[A, i IT} = 0 for every stopping time T. Since A is FV and predictable, by stopping at T - if necessary, we can assume the jumps of A are bounded. And since i is a Q local martingale, its supremum process is locally integrable (Theorem 34). This is enough to give us that [A, i J is Q locally of integrable variation. By stopping we assume it is integrable. We then have that A is Q natural, so if T is a stopping time, we replace A with AT and we still have a natural process, and thus we have EQ{[A, ilT} = O. Since Twas arbitrary, this implies that [A, il is a martingale. Of course, the process is implicitly stopped at several stopping times, so what we have actually shown is that [A, i 1 is a Q local martingale. We now have that 1 it 1 -zAt = Q-Iocal martingale + --dAs t 0 Zs- which in turn gives us, recalling that A = (X, Z), 1 1 X t = Zt (ZtXt) = Zt (Mt + (X, ZIt) 1 it 1 = -Mt + Q-Iocal martingale + -d(X, Z)s. Zt 0 Zs- Since Z (i M) = M = P local martingale, we have that ~M is also a Q local martingale. Thus finally X t - J~ z~_ d(X, Z)s is a Q local martingale, and the proof is done. 0 Let us next consider the interesting case where Q is absolutely continuous with respect to P, but the two probability measures are not equivalent. (That is, we no longer assume that P is also absolutely continuous with respect to Q.) We begin with a simple result, and we will assume F o = {0,n} a.s. Theorem 37. Let X be a P local martingale with Xo = O. Let Q be another probability absolutely continuous with respect to P, and let Zt = E {~IFt }. 8 Girsanov's Theorem 135 Assume that (X, Z) exists for P. Then At = f; z~_ d{X, Z)s exists a.s. for the probability Q, and X t - f; z~_ d{X, Z)s is a Q local martingale. Proof. Zo = E{Z} = 1, so if we let Rn = inf{t > 0 : Zt ~ lin}, then Rn increase to 00, Q-a.s., and the process z:_ is bounded on [0, Rn ]. By Theorem 36, Xt-n - f; z~_ d{X, Z):-n is a Q local martingale, each n. Since a local, local martingale is a local martingale, we are done. 0 We now turn to the general case, where we no longer assume the existence of (X, Z), calculated with P. (As before, we take X to be a P local martingale.) We begin by defining a key stopping time: R = inf{t > 0: Zt = 0, Zt- > O}. Note that Q(R < 00) = 0, but it is entirely possible that P(R < 00) > O. We further define Ut = ~XR1{t2:R}. Then U is an FV process, and moreover U is locally integrable (dP). Let Tn increase to 00 and be such that XTn is a uniformly integrable martingale. Then ~ ~ - Thus U has a compensator U, and of course U is predictable and U - U is a P local martingale. Theorem 38 (Lenglart-Girsanov Theorem). Let X be a P local martin- gale with Xo = O. Let Q be a probability absolutely continuous with respect to P, and let Zt = Ep{~IFt}, R = inf{t > 0 : Zt = 0, Zt- > O}, and Ut = ~XR1{t2:R}' Then i t 1X t - -d[X, Z]s + Uto Zs is a Q local martingale. Proof. Let Rn = inf{t > 0 : Zt ~ ~}. (Recall that Zo = 1, and also note that it is possible that Rn = R.) Then both X R n and Z R n are P local R t ~ martingales. Also note that At n = fo zfn l{z.;zn>o}d[XRn,ZRn]s, URn, and yRn = XRn - ARn + URn are all P-well-defined. We can define At = I t ~s l{zs>o}d[X, Z]s on [O,R), since d[XRn,ZRn]s does not charge (R,oo), and zfn l{z;Zn>o} = 0 at R. Thus we need only to show Y R n is a Q local martingale for each fixed n, which is the same as showing that ZRnyRn is a P local martingale. Let us assume all these processes are stopped at Rn to simplify notation. We have ZY=ZX-ZA+ZU. 136 III Semimartingales and Decomposable Processes Hence, d(ZX) = Z~dX + X_dZ + d[Z, X] = local martingale + d[Z, X] d(AZ) = A_dZ + ZdA = local martingale + ZdA = local martingale + l{z>o}d[X, Z] d(ZU) = Z_dU + UdZ local martingale + Z_dU = local martingale + Z_dU where the last equality uses that U - Uis a local martingale (dP). Summarizing we have ZY=ZX-ZA+ZU = local martingale + [Z, X] - local martingale + Jl{z>o}d[X, Z]) + local martingale +JZ_dU which we want to be a local martingale under dP. This will certainly be the case if However (*) equals ~ZR~XRl{tf\Rn2:R}+ ZR-~XRl{tf\Rn2:R}. But ~ZR = ZR-ZR- = O-ZR_ = -ZR-, and this implies that equation (*) is indeed zero, and thus the Lenglart-Girsanov Theorem holds. 0 Corollary. Let X be a continuous local martingale under P. Let Q be abso- lutely continuous with respect to P. Then (X, Z) = [Z, X] = [Z, X]C exists, and X t - t Zl d[Z,X]~ = X t - rt Dsd[X,X]sJo s- Jo which is a Q local martingale. Proof By the Kunita-Watanabe inequality we have which shows that it is absolutely continuous with respect to d[X, X]s a.s., whence the result. 0 8 Girsanov's Theorem 137 We remark that if Z is the solution of a stochastic exponential equation of the form dZs = Zs_HsdXs (which it often is), then as = H s. Example. A problem that arises often in mathematical finance theory is that one has a semimartingale S = M +A defined on a filtered probability space (0, F, IF, P) satisfying the usual hypotheses, and one wants to find an equivalent probability measure Q such that under Q the semimartingale X is a local martingale, or better, a martingale. In essence this amounts to finding a probability measure that "removes the drift." To be concrete, let us suppose S is the solution of a stochastic differential equation3 dSs = h(s, Ss)dBs + b(s; Sri r ::; s)ds, where B is a standard Wiener process (Brownian motion) under P. Let us postulate the existence of a Q and let Z = ~~ and Zt = E{ZIFtl, which is clearly a cadlag martingale. By Girsanov's Theorem rt rt 1 rJo h(s, Ss)dBs - Jo Zs d[Z, Joh(r, Sr)dBrJs is a Q local martingale. We want to find the martingale Z. In Chap. IV we will study martingale representation and show in particular that every local martingale on a Brownian space is a stochastic integral with respect to Brownian motion. Thus we can write Zt = 1 + J~ JsdBs for some predictable process J. If we assume Z is well behaved enough to define H s = t, then we have Zt = 1 + J~ H sZsdBs, which gives us a linear stochastic differential equation to solve for Z. Thus if we let Nt = J~ HsdBs, we get that Zt = £(N)t. 4 It remains to determine H. We do this by observing from our previous Girsanov calculation that rt h(s, Ss)dBs _ t J..-ZsHsh(s, Ss)ds = r t h(s, Ss)dBs _ r t Hsh(s, Ss)dsJo Jo Zs Jo Jo Q I I . I ur h h H -b(s; Sri r < s) h' h . ldis a oca martmga e. vve t en c oose s = h(s, Ss)- ,w IC yIe S St= l\(S,Ss)dBs + l\(s;Sr;r::;s)ds rt b(s· S . r < s) is a local martingale under Q. Letting Mt = Bt + Jo ~(:: S:; ds denote this Q local martingale, we get that [M, Mlt = [B, Blt = t, and by Levy's Theorem M is a Q-Brownian motion. Finally, under Q we have that S satisfies the stochastic differential equation 3 Stochastic differential equations are introduced and studied in some detail in Chap. V. 4 The stochastic exponential [ is defined on page 85. 138 III Semimartingales and Decomposable Processes There is one problem with the preceding example: we do not know a priori whether our solution Z is the Radon-Nikodym density for simply a measure Q, or whether Q is an actual bona fide probability measure. This is a constant problem. Put more formally, we wish to address this problem: Let M be a local martingale. When is £ (M) a martingale? The only known general conditions that solve this problem are Kazamaki's criterion and Novikov's criterion.5 Moreover, these criteria apply only to lo- cal martingales with continuous paths. Novikov's is a little less powerful than Kazamaki's, but it is much easier to check in practice. Since Novikov's crite- rion follows easily from Kazamaki's, we present both criteria here. Note that if M is a continuous local martingale, then of course £ (M) is also a continuous local martingale. Even if, however, it is a uniformly integrable local martin- gale, it still need not be a martingale; we need a stronger condition. As an example, one can take u(x) = Ilxll- 1 and Nt = u(Bd where B is standard three dimensional Brownian motion. Then N is a uniformly integrable local martingale but not a martingale. Nevertheless, whenever M is a continuous local martingale, then £(M) is a positive supermartingale, as is the case with any nonnegative local martingale. Since £(M)o = 1, and since £(M) is a positive supermartingale, we have E{£(M)tl :::; 1, all t. (See the lemma below.) It is easy to see that Z is a true martingale if one also has E{£(M)tl = 1, for all t. We begin with some preliminary results. Lemma. Let M be a continuous local martingale with M o = O. Then E{£(M)t} :::; 1 for all t 2 O. Proof. Recall that £(M)o = 1. Since M is a local martingale, £(M) is a nonnegative local martingale. Let Tn be a sequence of stopping times reducing £(M). Then E{£(M)tAT n } = 1, and using Fatou's Lemma, E{£(M)tl = E{liminf £(MhAT n } :::; liminf E{£(MhAT n } = 1. 0 n--+oo n--+oo Theorem 39. Let M be a continuous local martingale. Then E{e!Mt } :::; E{e![M,Mlt}1/2. Proof (£(M))! = (eMt-![M,Mlt)! = e!M'(e-![M,Ml,)! which implies that 5 There also exist partial results when the local martingale is no longer continuous but only has cadlag paths. See Exercise 14 of Chap. V for an example of these results. 8 Girsanov's Theorem 139 e~Mt = (£(M)t)~ (e~[M,Mjt)~ and this together with the Cauchy-Schwarz inequality E{£(M)tl ::; 1 gives the result. and the fact that o Lemma. Let M be a continuous local martingale. Let 1 < p < 00, 1 + 1 = 1. Taking the supremum below over all bounded stopping times, assu~e £hat supE{e(2J,[1)Mr} < 00. T Then £(M) is an Lq bounded martingale. Proof. Let 1 < p < 00 and r = ~~~. Then s = ~+l and ~ + ~ = 1. Also we note that (q - J;)s = 2(Jl--l) which we use in the last equality of the proof. We have £(M) = eqM- 1[M,M] = e.;rM-1[M,M]e(q-.;r)M We now apply Holder's inequality for a stopping time S: E{£(M)~} = E{evqrMs-Â¥[M,Mls}~ E{e(q-.;r)Ms}~ = (E{£(y'qrM)s})~ (E{ e( 2($-1» Ms})~ Recalling that E{£(y'qrM)s} ::; 1, we have the result. o Theorem 40 (Kazamaki's Criterion). Let M be a continuous local mar- tingale. Suppose suPTE{eC!MT)} < 00, where the supremum is taken over all bounded stopping times. Then £(M) is a uniformly integrable martingale. Proof. Let 0 < a < 1, and p > 1 be such that (.Jf!.-l) < ~. Our hypothesis combined with the preceding lemma imply that £(aM) is an Lq bounded martingale, where ~ + ~ = 1, which in turn implies it is a uniformly integrable martingale. However £(aM) = eaM-4[M,Mj = ea2M-a22 [M,M]ea(l-a)M = £(Mt2ea(l-a)M, and using Holder's inequality with a-2 and (1 - a2 )-1 yields (where the 1 on the left side comes from the uniform integrability): 140 III Semimartingales and Decomposable Processes Now let a increase to 1 and the second term on the right side of the last inequality above converges to 1 since 2a(1 - a) ----+ o. Thus 1 ::; E{£(M)oo}, and since we know that it is always true that 1 ~ E{£(M)oo}, we are done. 0 As a corollary we get the very useful Novikov's criterion. Because of its importance, we call it a theorem. E{e![M,M]oo} < 00. Then £(M) is a uniformly integrable martingale. Theorem 41 (Novikov's Criterion). Let M be a continuous local martin- gale, and suppose that Proof By Theorem 39 we have E{d MT } ::; E{d[M,Mlr}!, and we need only to apply Kazamaki's criterion (Theorem 40). 0 We remark that it can be shown that 1/2 is the best possible constant in Novikov's criterion, even though Kazamaki's criterion is slightly stronger. Note that in the case of the example treated earlier, we have [N, Nlt = J~ H;ds h H b(s'S -r 8 Girsanov's Theorem 141 where W is a standard Wiener process and where St is thought of as "I; x(s)ds," the cumulative received signal. The key step in our analysis is the following consequence of the Girsanov- Meyer Theorem. Theorem 42. Let W be a standard Brownian motion on (O,F,lF,P), and let HElL be bounded. Let X t = it Hsds + Wt and define Q by 'Â¥j = exp{J(; -HsdWs - ~ 1(; H;ds}, for some T > o. Then under Q, X is a standard Brownian motion for 0::; t ::; T. Proof Let ZT = exp{Jt -HsdWs -! It H;ds}. Then if Zt = E{ZTIFtl we know by Theorem 37 of Chap. II that Z satisfies the equation Zt = 1 -it Zs_HsdWs. By the Girsanov-Meyer Theorem (Theorem 35), we know that Nt = Wt -it ;s d[Z, W]s is a Q local martingale. However [Z, W]t = [-Z_H. w, Wlt = it -ZsHsd[W, W]s = -it ZsHsds, since [W, Wlt = t for Brownian motion. Therefore Nt = Wt - r t -~ZsHsdsJo Zs = Wt +it Hsds =xt , hence X is a Q local martingale. Since (J~ Hsdsk?o is a continuous FV pro- cess we have that [X, X]t = [W, Wlt = t, and by Levy's Theorem (Theorem 39 of Chap. II) we conclude that X is a standard Brownian motion. 0 Corollary. Let W be a standard Brownian motion and HElL be bounded. Then the law of X t = it Hsds + wt , o::; t ::; T < 00, is equivalent to Wiener measure. 142 III Semimartingales and Decomposable Processes Proof Let qo, T] be the space of continuous functions on [0, T] with values in lR (such a space is called a path space). If W = (Wt )O:5;t:5;T is a standard Brownian motion, it induces a measure /-Lw on e[O, T]: /-Lw(A) = P{w : t f--+ Wt(w) E A}. Let /-Lx be the analogous measure induced by X. Then by Theorem 42 we have /-Lx rv /-LW and further we have d/-LW { rT 1 rT 2 }d/-LX = exp J o -HsdWs - 2" Jo Hsds . o reject H o, fail to reject H o, Remark. We have not tried for maximum generality here. For example the hypothesis that H be bounded can be weakened. It is also desirable to weaken the restriction that H E JL. Indeed we only needed that hypothesis to be able to form the integral I~ HsdWs. This is one example to indicate why we need a space of integrands more general than JL. We are now in a position to consider the problem posed earlier: is there a signal corrupted by noise, or is there just noise (that is, does s(t) = °a.e., a.s.)? In terms of hypothesis testing, let H o denote the null hypothesis, HI the alternative. We have: H o : X T = WT T HI : X T = 1Hsds + WT. We then have ~: = exp (iT -HsdWs - ~ iT H;dS) , by the preceding corollary. This leads to a likelihood ratio test: if dd/-LW (w) ::; A, /-Lx if dd/-LW (w) > A, /-Lx where the threshold level A is chosen so that the fixed Type I error is achieved. To indicate another use of the Girsanov-Meyer Theorem let us consider stochastic differential equations. Since stochastic differential equations6 are treated systematically in Chap. V we are free here to restrict our attention to a simple but illustrative situation. Let W be a standard Brownian motion on a space (O,F,lF,P) satisfying the usual hypotheses. Let Ii(w,s,x) be functions satisfying (i = 1,2): 6 Stochastic "differential" equations have meaning only if they are interpreted as stochastic integral equations. 9 The Bichteler-Dellacherie Theorem 143 (i) Ifi(W, S, x) -li(w, s, y)1 :::; Klx - yl for fixed (w, s); (ii) Ii("s,x) E F s for fixed (s,x); (iii) Ii (w, ., x) is left continuous with right limits for fixed (w, x). By a Picard-type iteration procedure one can show there exists a unique so- lution (with continuous paths) of X t = X o +it h(·,s,Xs)dWs + it h(·,s,Xs)ds. The Girsanov-Meyer Theorem allows us to establish the existence of solutions of analogous equations where the Lipschitz hypothesis on the "drift coeffi- cient" h is removed. Indeed if X is the solution of (* * *), let --y be any bounded, measurable function such that --y(w, s, X s) E lL. Define g(w, s,x) = h(w, s,x) +h(w, s,xh(s,w,x). We will see that we can find a solution of provided we choose a new Brownian motion B appropriately. We define a new probability law Q by dQ _ ( rT 1 rT 2)dP-exp Jo --y(s,Xs)dWs -2"Jo --y(s,Xs)ds . By Theorem 42 we have that B t = W t -it --y(s,Xs)ds is a standard Brownian motion under Q. We then have that the solution X of (* * *) also satisfies Xt=Xo +I t !lc,s,Xs)dBs + I t (h+fnK,s,Xs)ds = X o + it h(·,s,Xs)dBs +it g(',s,Xs)ds, which is a solution of a stochastic differential equation driven by a Brownian motion, under the law Q. 9 The Bichteler-Dellacherie Theorem In Sect. 7 we saw that a classical semimartingale is a semimartingale. In this section we will show the converse. 144 III Sernimartingales and Decomposable Processes Theorem 43 (Bichteler-Dellacherie Theorem). An adapted, cadlag pro- cess X is a semimartingale if and only if it is a classical semimartingale. That is, X is a semimartingale if and only if it can be written X = M + A, where M is a local martingale and A is an FV process. Proof. The sufficiency is exactly Theorem 26. We therefore establish here only the necessity. Since X is dtdlag, the process Jt = LO 0, and by the above it suffices to show X is a classical semimartingale on [0, uo]. Thus it is no loss to assume X is a total semimartin- gale on [0, uo]. We will show that X is a quasimartingale, under an equivalent probability Q. Roo's Theorem (Theorem 15) shows that X is a classical semi- martingale under Q, and the Girsanov-Meyer Theorem (Theorem 35) then shows that X is a classical semimartingale under P. Let us take H E S of the special form: n-l Ht = L Hi1CTi,Ti+l] i=O where °= To :::; Tl :::; ... :::; Tn-l < Tn = uo. In this case the mapping Ix is given by Ix(H) = (H. X)uo = HO(XT1 - X o) + ... + Hn - 1(Xuo - X Tn _1)· The mapping Ix : Su ----+ L O is continuous, where LO is endowed with the topology of convergence in probability, by the hypothesis that X is a total semimartingale. Let B = {H E S: H has a representation Hand IHI :::;-l}. Let f3 = Ix (B), the image of B under Ix. It will now suffice to find a probability Q equivalent to P such that X t E L1(dQ), ° :::; t :::; Uo and such that sUPUE{3 EQ(U) = c < 00. The reason this suffices is that if we take, for a given ° = to < tl < ... < tn = Uo, the random variables 9 The Bichteler-Dellacherie Theorem 145 H o = sign(EQ{Xtl - XoIFo}), HI = sign(EQ{Xt2 - XiI IFtJ), ... , we have that for this H E B, Since this partition T was arbitrary, we have Var(X) = sUPr Varr(X) ~ sUPUE{3 EQ(U) = c < 00, and so X is a Q quasimartingale. Lemma 1. lim sup P(IYI > c) = o. c--+oo YE{3 Proof of Lemma 1. Suppose limc--+oo sUPYE{3 P(IYI > c) > O. Then there ex- ists a sequence en tending to 00, Yn E (3, and a> 0 such that P(lYnl > en) :::: a, all n. This is equivalent to P C~I > 1) :::: a > O. Since Yn E (3, there exists Hn E B such that Ix(Hn) = Yn. Then Ix(c: Hn) = -lIx(Hn) = -lYn E (3, if Cn :::: 1. But -lHn tends to 0 uniformly a.s. whichen en en implies that Ix(...!...Hn) = -lYn tends to 0 in probability. This contradictsen en (*). 0 Lemma 2. There exists a law Q equivalent to P such that X t E L 1(dQ), o~ t ~ uo. Proof of Lemma 2. Let Y = sUPo 146 III Semimartingales and Decomposable Processes Let A E :F such that Q(A) > O. Then there exists a constant d such that Q(( > d) ::; Q(A)/2, for all ( E (3, by assumption. Using this constant d, let c = 2d, and we have that 0 ::; cIA tt (3, and moreover if B+ denotes all bounded, positive r.v., then cIA is not in the L 1 (dQ) closure of (3-B+, denoted (3 - B+. That is, cIA tt (3 - B+. Since the dual of L1 is Loo, and (3 - B+ is convex, by a version of the Hahn-Banach Theorem (see, e.g., Treves [223, page 190]) there exists a bounded random variable Y such that Replacing TJ by al{y Exercises for Chapter III 147 Bibliographic Notes The material of Chap. III comprises a large part of the core of the "general theory of processes" as presented, for example in Dellacherie [41]' or alterna- tively Dellacherie and Meyer [45, 46, 44]. We have tried once again to keep the proofs non-technical, but instead of relying on the concept of a natural process as we did in the first edition, we have used the approach of R. Bass [11], which uses the P. A. Meyer classification of stopping times and Doob's quadratic in- equality to prove the Doob-Meyer Theorem (Theorem 13), the key result of the whole theory. The Doob decomposition is from Doob [55], and the Doob-Meyer decom- position (Theorem 13) is originally due to Meyer [163, 164]. The theory of quasimartingales was developed by Fisk [73], Orey [187], K. M. Roo [207], Stricker [218], and Metivier-Pellaumail [159J. The treatment of compensators is new to this edition. The simple example of the compensator of a process with one jump of size one (Theorem 17), dates back to 1970 with the now classic paper of Dellacherie [40J. The case of many jumps (Theorem 19) is due to C. S. Chou and P. A. Meyer [30], and can be found in many texts on point processes, such as [24] or [139]. The example of hazard rates comes from Fleming and Harrington [74]. The Fundamental Theorem of Local Martingales is due to J. A. Yan and appears in an article of Meyer [172]; it was also proved independently by Doleans-Dade [51J. Le Jan's Theorem is from [112J. The notion of special semimartingales and canonical decompositions is due to Meyer [171]; see also Yoeurp [234]. The Girsanov-Meyer theorems (The- orems 35 and 36) trace their origin to the 1954 work of Maruyama [152], followed by Girsanov [83], who considered the Brownian case only. The two versions presented here are due to Meyer [171], and the cases where the two measures are not equivalent (Theorems 37 and 38) are due to Lenglart [142J. The example from finance theory (starting on page 137) is inspired by [203J. Kazamaki's criterion was published originally in 1977 [124J; see also [125], whereas Novikov's condition dates to 1972 [184J. The Bichteler-Dellacherie Theorem (Theorem 43) is due independently to Bichteler [13, 14] and Dellacherie [42]. It was proved in the late 1970's, but the first time it appeared in print was when J. Jacod included it in his 1979 tome [103]. Many people have made contributions to this theorem, which had at least some of its origins in the works of Metivier-Pellaumail, Mokobodzki, Nikishin, Letta, and Lenglart. Our treatment was inspired by Meyer [176J and by Yan [233]. Exercises for Chapter III Exercise 1. Show that the maximum and the minimum of a finite number of predictable stopping times is still a predictable stopping time. 148 Exercises for Chapter III Exercise 2. Let S be a totally inaccessible stopping time, and let T = S + 1. Show that T is a predictable stopping time. Exercise 3. Let S, T be predictable stopping times. Let A = {S = T}. Show that SA is a predictable stopping time. Exercise 4. Let (0, F, IF, P) be a filtered probability space satisfying the usual hypotheses. Show that the predictable a-algebra (on JR.+ x 0) is con- tained in the optional a-algebra. (Hint: Show that a dldlag, adapted process can be approximated by processes in lL.) Exercise 5. Let S, T be stopping times with S :s: T. Show that (S, T] = ((t,w): S(w):s: t:s: T(w)} is a predictable set. Show further that [S,T) and (S, T) are optional sets. Last, show that if T is predictable, then (S, T) is a predictable set, and if both Sand T are predictable, then [S, T) is a predictable set. Exercise 6. Let T be a predictable stopping time, and let (Sn)n>l be a sequence of stopping times announcing T. Show that FT- = Vn FS n ~ Exercise 7. A filtration IF is called quasi left continuous if for every pre- dictable stopping time T one has FT = FT _.7 Show that if IF is a quasi left continuous filtration, then whenever one has a non-decreasing sequence of stopping times (Sn)n~l' with limn-+oo Sn = T, it follows that FT = Vn Fsn • (Note: One can have a quasi left continuous filtration such that there exists a stopping time T with FT_ -I- FT.) Exercise 8. Show that the natural completed filtration of a Levy process is a quasi left continuous filtration. *Exercise 9. Let X be a semimartingale with E{[X, X]oo} < 00 and suppose X has a decomposition X = X o + M + A, with A predictable. Show that E{[A, A]oo} :s: E{[X, X]oo}. (Hint: Recall that if T is a predictable stopping time, then 6.AT1{T Exercises for Chapter III 149 Exercise 11. Show that if A is a predictable finite variation process, then it is of locally integrable variation. That is, show that there exists a sequence of stopping times (Tnk:::l such that E{foTn IdAsl} < 00. Exercise 12. Let N be a counting process with its minimal completed fil- tration, with independent interarrival times. Show that the compensator A of N has absolutely continuous paths if and only if the cumulative distribution functions of the interarrival times are absolutely continuous. Exercise 13. Let T be an exponential random variable and Nt = 1{t2T}. Let Fo = a{T} and suppose the filtration IF is constant: Ft = Fo for all t ::::: 0. Show that the compensator A of N is At = 1{t2T }. (This illustrates the importance of the filtration when calculating the compensator.) Exercise 14. Let N be a compound Poisson process, Nt = Li>l Ui l{t2T;} where the times Ti are the arrival times of a standard Poisson process with parameter A and the Ui are i.i.d. and independent of the arrival times. Suppose E{IUil} < 00 and E{Ui } = J1. Show that the compensator A of N for the natural filtration is given by At = AJ1t. Exercise 15. Show that if T is exponential with parameter A, its hazard rate is constant and equal to A. Show also that if R is Weibull with parameters (0:, (3), then its hazard rate is A(t) = o:(3C>tc>-l. Exercise 16. Let T be exponential with parameter A and have joint distri- bution with U given by P(T > t, U > s) = exp{ -At - J1S - Bts} for t :::: 0, s ::::: 0, where A, J1, and B are all positive constants and also B :::; AJ1. Show that the crude hazard rate of (T, U) is given by A#(t) = A+ Bt. *Exercise 17. Let M be a martingale on a filtered space where the filtra- tion is quasi left continuous. Show that (M, M) is continuous. (Hint: See the discussion on quasi left continuous filtrations on page 189 of Chap. IV.) Exercise 18. Let X be a semimartingale such that the process D t = sUPsl of stopping times increasing to 00 a.s. such that XTn = Mn + An, with An a predictable finite variation process, for each n. Show that X is special. (That is, a semimartingale which is locally special is also special.) Exercise 20. Prove the Lemma on page 132. Let Q rv P, and Zt = Ep {~~ 1Ft}. Show that an adapted, dldlag process M is a Q local martingale if and only if M Z is a P local martingale. *Exercise 21. Let IF c G be filtrations satisfying the usual hypotheses, and let X be a G quasimartingale. Suppose that yt = E{XtIFtl, and that Y can be taken cadlag (this can be proved to be true, although it is a little hard to 150 Exercises for Chapter III prove). Show that Y is also a quasimartingale for the filtration JF. (Hint: Use Rao's Theorem.) *Exercise 22. Suppose that A is a predictable finite variation process with 1 E{[A, All} < 00, and that M is a bounded martingale. Show that [A, Ml is a uniformly integrable martingale. Exercise 23. Let T be a strictly positive random variable, and let :Pi = O'{T 1\ s; s :s: t}. Show that (Ftk::o is the smallest filtration making T a stopping time. Exercise 24. Let Z be a cadlag supermartingale of Class D with Zo = 0 and suppose for all predictable stopping times T one has E{~ZTIFT-} = 0, a.s. Show that if Z = M - A is the unique Doob-Meyer decomposition of Z, then A has continuous paths almost surely. Exercise 25. Let A be an increasing process of integrable variation, and let T be a predictable jump time of A such that E{~ATIFT_} = o. Then its compensator A is continuous at T. (This exercise complements Theorem 21.) *Exercise 26. A supermartingale Z is said to be regular if whenever a sequence of stopping times (Tn)n~1 increases to T, then limn -+oo E{ ZTn } = E{ZT}. Let Z be a cadlag supermartingale of Class D with Doob-Meyer decomposition Z = M - A. Show that A is continuous if and only if Z is regular. *Exercise 27 (approximation of the compensator by Laplacians). Let Z be a dtdlag positive supermartingale of Class D with limt-+oo E{Zd = O. (Such a supermartingale is called a potential.) Let Z = M - A be its Doob- Meyer decomposition and assume further that A is continuous. Define Show that for any stopping time T, limh-+O A~ = AT with convergence in L 1 . *Exercise 28. Let Z be a dtdlag positive supermartingale of Class D with limt-+oo E{Zd = O. Let Z = M -A be its Doob-Meyer decomposition. Let Ah be as given in Exercise 27. Show that for any stopping time T, limh-+O A~ = AT, but in this case the convergence is weak for L 1 ; that is, the convergence is in the topology O'(L 1 , LOO).8 Exercise 29 (discrete Laplacian approximations). Let Z be a cadlag positive supermartingale of Class D with limt-+oo E{ Zd = O. Let Z = M - A be its Doob-Meyer decomposition and assume further that A is continuous. Define 8 x n converges to X in cr(L I , £C>O) if X n , X are in L 1 and for any a.s. bounded random variable Y, E(XnY) -+ E(XY). Exercises for Chapter III 151 00 A~=""E{Zk -Zk+lIFk}~ 2lt" ~ 2lt" k=O Show that limn -+ oo A~ = Aoo with convergence in L1 . Exercise 30. Use Meyer's Theorem (Theorem 4) to show that if X is a strong (Markov) Feller process for its natural completed filtration 1Ft-', and if X has continuous paths, then the filtration JFJL has no totally inaccessible stopping times. (This implies that the natural filtration of Brownian motion does not have any totally inaccessible stopping times.) *Exercise 31. Let (n, F, JF, P) be the standard Brownian space. Show that the optional a-algebra and the predictable a-algebra coincide. (Hint: Use Meyer's Theorem (Theorem 4) and Exercise 30.) *Exercise 32. Let (n, F, JF, P) be a filtered probability space satisfying the usual hypotheses. Let X be a (not necessarily adapted) cadlag stochas- tic process such that for A > 0, EU;' e-.-\tIXt!dt} < 00. Let R.-\(Xt} E{Jooo e- AS Xt+sdsIFd, the right continuous version. Show that is an JF martingale. *Exercise 33 (Knight's compensator calculation method). Let X be a cadlag semimartingale. In the framework of Exercise 32 suppose the limits below exist both pathwise a.s. and are in L 1 , and are of finite variation in finite time intervals: Show that X is a special semimartingale, and A is the predictable term in its semimartingale decomposition. IV General Stochastic Integration and Local Times 1 Introduction We defined a semimartingale as a "good integrator" in Chap. II, and this led naturally to defining the stochastic integral as a limit of sums. To express an integral as a limit of sums requires some path smoothness of the integrands and we limited our attention to processes in JL, the space of adapted processes with paths that are left continuous and have right limits. The space JL is sufficient to prove Ito's formula, the Girsanov-Meyer Theorem, and it also suffices in some applications such as stochastic differential equations. But other uses, such as martingale representation theory or local times, require a larger space of integrands. In this chapter we define stochastic integration for predictable processes. Our extension from Chap. II is very roughly analogous to how the Lebesgue integral extends the Riemann integral. We first define stochastic integration for bounded, predictable processes and a subclass of semimartingales known as 1{2. We then extend the definition to arbitrary semimartingales and to locally bounded predictable integrands. We also treat the issue of when a stochastic integral with respect to a martingale or a local martingale is still a local martingale, which is not always the case. In this respect we treat the subject of sigma martingales, which has recently been shown to be important for the theory of mathematical finance. 2 Stochastic Integration for Predictable Integrands In this section, we will weaken the restriction that an integrand H must be in JL. We will show our definition of stochastic integrals can be extended to a class of predictably measurable integrands. Throughout this section X will denote a semimartingale such that X o = O. This is a convenience involving no loss of generality. If Y is any semimartin- gale we can set Yt = yt - Yo, and if we have defined stochastic integrals for 154 IV General Stochastic Integration and Local Times semimartingales that are zero at 0, we can next define it HsdYs == it HsdYs + HoYo. When Yo =I- 0, recall that we write J~+ HsdYs to denote integration on (0, t], and J~ HsdYs denotes integration on the closed interval [0, t]. We recall for convenience the definition of the predictable a-algebra, al- ready defined in Chap. III. Definition. The predictable a-algebra P on JR.+ x n is the smallest a- algebra making all processes in lL measurable. That is, P = a{H : HElL}. We let bP denote bounded processes that are P measurable. Let X = M + A be a decomposition of a semimartingale X, with X o = Mo = Ao = O. Here M is a local martingale and A is an FV process (such a decomposition exists by the Bichteler-Dellacherie Theorem (Theorem 43 of Chap. III)). We will first consider special semimartingales. Recall that a semimartingale X is called special if it has a decomposition where N is a local martingale and A is a predictable FV process. This decom- position is unique by Theorem 30 in Chap. III and it is called the canonical decomposition. Definition. Let X be a special semimartingale with canonical decomposition X = N + A. The 712 norm of X is defined to be The space of semimartingales 1-e consists of all special semimartingales with finite 1{2 norm. In Chap. V we define an equivalent norm which we denote II . 11M;2. Theorem 1. The space of 712 semimartingales is a Banach space. Proof. The space is clearly a normed linear space and it is easy to check -2 -- that II . IIH2 is a norm (recall that E{N oo} = E{[N, NJoo}, and therefore IIXIIH2 = 0 implies that E{N~} = 0 which implies, since N is a martingale, that N == 0). To show completeness we treat the terms N and A separately. Consider first N. Since E{N~} = II[N,N]~21Ii2' it suffices to show that the space of L 2 martingales is complete. However an L 2 martingale M can be identified with Moo E L2, and thus the space is complete since L2 is complete. Next suppose (An) is a Cauchy sequence of predictable FV processes in II . 112 where IIAllp = II Jooo IdAsIII LP , P:::: 1. To show (An) converges it suffices 2 Stochastic Integration for Predictable Integrands 155 to show a subsequence converges. Therefore without loss of generality we can assume Ln IIAn ll 2 < 00. Then L An converges in II . III to a limit A. Moreover lim L roo IdA~1 = 0 m-+oo n2:m Jo in L 1 and is dominated in L 2 by Ln Jooo IdA~I. Therefore L An converges to the limit A in II . 112 as well, and there is a subsequence converging almost surely. To see that the limit A is predictable, note that since each term in the sequence (An)n2:1 is predictable, the limit A is the limit of predictably measurable processes and hence also predictable. 0 For convenience we recall here the definition of lL. Definition. lL (resp. blL) denotes the space of adapted processes with caglad1 (resp. bounded, caglad) paths. We first establish a useful technical lemma. Lemma. Let A be a predictable FV process, and let H be in lL such that E{fooo IHslldAsl} < 00. Then the FV process (J~ HsdAs)t2:o is also pre- dictable. Proof. We need only to write the integral J~ HsdAs as the limit of Riemann sums, each one of which is predictable, and which converge in ucp to J~ H s dAs , showing that it too is predictable. 0 The results that follow will enable us to extend the class of stochastic integrands from blL to bP, with X E r[2 (and X o = 0). First we observe that if H E blL and X E 1{2, then the stochastic integral H . X E 112 . Also if X = N + A is the canonical decomposition of X, then H . N + H . A is the canonical decomposition of H . X by the preceding lemma. Moreover, The key idea in extending our integral is to notice that [N, N] and A are FV processes, and therefore w-by-w the integrals J~ H;(w)d[N, N]s(w) and J~ IHslldAsl make sense for any H E bP and not just HEll. Definition. Let X E 112 with X = N + A its canonical decomposition, and let H, J E bP. We define dx(H, J) by 1 "caglad" is the French acronym for left continuous with right limits. 156 IV General Stochastic Integration and Local Times Theorem 2. For X E 1{2 the space bJL is dense in bP under dx (-, .). Proof. We use the Monotone Class Theorem. Define A = {H E bP: for any c > 0, there exists J E bJL such that dx(H,J) < c}. Trivially A contains bJL. If H n E A and H n increases to H with H bounded, then H E bP, and by the Dominated Convergence Theorem if J > °then for some N(J), n > N(J) implies dx(H, H n ) < J. Since each H n E A, we choose no > N(J) and there exists J E bJL such that dx(J, Hno ) < J. Therefore given c > 0, by taking J = c/2 we can find J E bJL such that dx(J, H) < c, and therefore H E A. An application of the Monotone Class Theorem yields the result. 0 Theorem 3. Let X E 1{2 and H n E bJL such that Hn is Cauchy under dx . Then H n . X is Cauchy in 1{2. Proof. Since IIHn.X -Hm'XII'H2 = dx(Hn , H m), the theorem is immediate. o Theorem 4. Let X E 1{2 and H E bP. Suppose H n E bJL and Jm E bJL are two sequences such that limn dx(Hn , H) = limmdx(Jm,H) = 0. Then Hn . X and Jm . X tend to the same limit in 1{2. Proof Let Y = limn-+ oo Hn . X and Z = limm-+oo Jm . X, where the limits are taken in 1{2. For £ > 0, by taking nand m large enough we have IIY - ZII'H2:s: IIY - Hn . XII'H2+ IIHn . X - Jm. Xllw + IIJm . X - ZII'H2 :s: 2c + IIHn . X - Jm . XII'H2 :s: 2£ + dx(Hn , Jm) :s: 2£ + dx(Hn , H) + dx(H, Jm) :s: 4£, and the result follows. o We are now in a position to define the stochastic integral for H E bP (and X E 1{2). Definition. Let X be a semimartingale in 1{2 and let H E bP. Let H n E bJL be such that limn-+oo dx(Hn,H) = 0. The stochastic integral H· X is the (unique) semimartingale Y E 1{2 such that limn-+ oo H n . X = Y in 1{2. We write H . X = (J~ HsdXsk:::.o, We have defined our stochastic integral for predictable integrands and semimartingales in 1{2 as limits of our (previously defined) stochastic integrals. In order to investigate the properties of this more general integral, we need to have approximations converging uniformly. The next theorem and its corollary give us this. 2 Stochastic Integration for Predictable Integrands 157 Theorem 5. Let X be a semimartingale in H2. Then Proof For a process H, let H* = SUPt IHtl. Let X = N +A be the canonical decomposition of X. Then Doob's maximal quadratic inequality (Theorem 20 of Chap. I) yields and using (a + b)2 ::;. 2a2+ 2b2 we have E{(X*)2} ::; 2E{(N*)2} + 2E{(100 IdAs l)2} ::; 8E{[N, N]oo} + 211100 IdAs llli2 ::; 81IXII~2. o Corollary. Let (xn) be a sequence of semimartingales converging to X in H2. Then there exists a subsequence (nk) such that limnk->oo(Xnk -X)* = 0 a.s. Proof By Theorem 5 we know that (xn - X)* = SUPt IXI' - Xtl converges to 0 in £2. Therefore there exists a subsequence converging a.s. 0 We next investigate some of the properties of this generalized stochastic integral. Almost all of the properties established in Chap. II (Sect. 5) still hold. 2 Theorem 6. Let X, Y E H2 and H, K E bP. Then (H + K) . X = H . X + K . X, and H· (X + Y) = H . X + H . Y. Proof One need only check that it is possible to take a sequence Hn E bL that approximates H in both dx and dy. 0 Theorem 7. Let T be a stopping time. Then (H· xf = Hl[o,T] . X = H. (XT ). 2 Indeed, it is an open question whether or not Theorem 16 of Chap. II extends to integrands in bP. See the discussion at the end of this section. 158 IV General Stochastic Integration and Local Times Proof. Note that l[o,Tj E blL, so H1[o,Tj E bP. Also, X T is clearly still in H 2. Since we know this result is true for H E blL (Theorem 12 of Chap. II), the result follows by uniform approximation, using the corollary of Theorem 5. 0 Theorem 8. The jump process (tl(H . X)sk:~o is indistinguishable from (Hs(tlXs))s~o. Proof. Recall that for a process J, tlJt = Jt - Jt-, the jump of J at time t. (Note that H· X and X are dtdUtg semimartingales, so Theorem 8 makes sense.) By Theorem 13 of Chap. II we know the result is true for H E blL. Let H E bP, and let Hn E blL such that limn--->oo dx (Hn, H) = O. By the corollary of Theorem 5, there exists a subsequence (nk) such that lim (Hnk . X - H . X) * = 0 a.s. nk-+oo This implies that, considered as processes, lim ti.(Hnk. X) = ti.(H· X), nk-+OO outside of an evanescent set. 3 Since each Hnk E blL, we have tl(Hnk . X) = Hnk (tlX), outside of another evanescent set. Combining these, we have lim Hnk(tlX)l{6.x~o} = lim ti.(Hnk. X)l{6.x~o} nk -+00 nk-+oo = ti.(H . X)l{6.x~o}, and therefore In particular, the above implies that limnk--->oo Hf'k(W) exists for all (t,w) in {tlX =I- O}, a.s. We next form A={w: there exists t>O such that lim Hf'k(W)=l-Ht(w) and tlXt(w)=l-O}. nk-+OO Suppose P(A) > O. Then dx(Hnk,H) ~ 111A{ [oo(H';k -Hs)2d( L (tlNu )2HI1L2io O 00, and the right side of (*) does not. Therefore P(A) = 0, and we conclude tl(H· X) = limnk--->ooHnktlX = HtlX. 0 3 A set A C JR+ x n is evanescent if lA is a process that is indistinguishable from the zero process. 2 Stochastic Integration for Predictable Integrands 159 Corollary. Let X E Je, H E bP, and T a finite stopping time. Then H· (XT -) = (H· xf-. Proof By Theorem 8, (H· xf- = (H. xf - HTl:1XTl{t~T}' On the other hand, X T- = X T - l:1XTl{t~T}' Let At = l:1XTl{t~T}' By the bilinearity (Theorem 6), H· (XT-) = H· (XT) - H . A. Since H· (XT ) = (H· xf by Theorem 7, and H . A = HTt:.XTl{t~T}, the result follows. 0 The next three theorems all involve the same simple proofs. The result is known to be true for processes in bIL; let (Hn) E bIL approximate HE bP in dx(-, .), and by the corollary of Theorem 5 let nk be a subsequence such that lim (Hnk . X - H . X) * = 0 a.s. nk-+ oo Then use the uniform convergence to obtain the desired result. We state these theorems, therefore, without proofs. Theorem 9. Let X E H 2 have paths of finite variation on compacts, and H E bP. Then H . X agrees with a path-by-path Lebesgue-Stieltjes integral. Theorem 10 (Associativity). Let X E H2 and H,K E bP. Then K· X E H2 and H . (K . X) = (HK) . X. Theorem 11. Let X E H 2 be a (square integrable) martingale, and HE bP. Then H . X is a square integrable martingale. Theorem 12. Let X, Y E H 2 and H, K E bP. Then and in particular Proof As in the proof of Theorem 29 of Chap. II, it suffices to show [H· X, Y]t = it Hsd[X, Y]s' Let (Hn ) E bIL such that limn --+ oo dx(Hn, H) = O. Let T m = inf{t > 0: 1Y't1 > m}. Then (rm) are stopping times increasing to 00 a.s. and IY.'!'" I ~ m. 4 Since it suffices to show the result holds on [0, T m ), each m, we can assume without loss of generality that y_ is in bIL. Moreover, the Dominated Convergence Theorem gives limn --+ oo dx (Hny_, HY_) = O. By Theorem 29 of Chap. II, we have 4 Recall that Y_ denotes the left continuous version of Y. 160 IV General Stochastic Integration and Local Times [Hn . X, Y]t = it H-;d[X, Y]s and again by dominated convergence lim [Hn . X, Y] = {t Hsd[X, Y]s n-+oo io (t ~ 0), (t ~ 0). It remains only to show limn->oo[Hn . X, Y] = [H· X, Y]. Let zn = Hn . X, and let nk be a subsequence such that limnk->oo(Znk - Z)* = 0 a.s., where Z = H . X (by the corollary to Theorem 5). Integration by parts yields [znk, Y] = ZnkY - (L) . Znk _ (Z~k) . Y = znkY _ (Y_Hnk). X - (Z~k). Y, where we have used associativity (Theorem 10). We take limits so that lim [znk,Y] = ZY - L . (H· X) - Z_ . Y nk-+ OO = ZY - L . (Z) - Z_ . Y = [Z,Y] = [H .X,Y]. o At this point the reader may wonder how to calculate in practice a canon- ical decomposition of a semimartingale X in order to verify that X E 1{2. Fortunately Theorem 13 will show that 1{2 is merely a mathematical conve- nience. Lemma. Let A be an FV process with Ao = 0 and Jooo IdAsl E L 2 . Then A E 1{2. Moreover IIAII7-l2 ~ 611 Jooo IdAsIII L 2. Proof If we can prove the result for A increasing then the general result will follow by decomposing A = A+ - A-. Therefore we assume without loss of generality that A is increasing. Hence as we noted in Sect. 5 of Chap. III, the compensator A of A is also increasing and E{Aoo } = E{Aoo } < 00. Let M be a martingale bounded by a constant k. Since A - A is a local martingale, Corollary 2 to Theorem 27 of Chap. II shows that L = M(A - A) - [M, A - A] is a local martingale. Moreover Therefore L is a uniformly integrable martingale (Theorem 51 of Chap. I) and E{L oo } = E{Lo} = O. Hence 2 Stochastic Integration for Predictable Integrands 161 E{Moe(A - A)oe} = E{[M,A - A]oe} = E{[M, A]oe} - E{[M, A]oe} = E{[M, A]oe}, because A is natural. By the Kunita-Watanabe inequality (the corollary to Theorem 25 of Chap. II) E{I[M,A]oel} :::; (E{[M, M]oe}E{[A,A]oe} )1/2 1 1 :::; 2E {[M,M]oe} + 2 E{[A,A]oe}, where the second inequality uses 2ab :::; a2 + b2 . However E{[M,M]oe} = E{M~} (Corollary 4 of Theorem 27 of Chap. II) and also [A, A]oe :::; A~ a.s. Therefore { ~ 1 2 1 2E Moe(A - A)oe} :::; 2 E {Moe } + 2E{Aoe}. Since M is an arbitrary bounded martingale we are free to choose and we obtain and using the Monotone Convergence Theorem we conclude E{(A - A)~}:::; E{A~}. Consequently E{A~,} :::; 2E{A~} + 2E{(A - A)~} :::; 4E{A~} < 00, and A - A is a square integrable martingale, and for A increasing. o Remarks. The constant 6 can be improved to 1+V8 :::; 4 by not decomposing A into A+ and A-. This lemma can also be proved using the Burkholder- Gundy inequalities (see Meyer [171, page 347]). 162 IV General Stochastic Integration and Local Times In Chap. V we use an alternative nOrm for semimartingales which we denote II . IIHP, 1 ~ P < 00. The preceding lemma shows that the norms II . 117-£2 and if· IIH2 are equivalent. The restrictio~s of integrands to bP and semimartingales to 1t2 are mathematically convenient but not necessary. A standard method of re- laxing such hypothesis is to consider cases where they hold locally. Recall from Sect. 6 of Chap. I that a property 1f is said to hold locally for a process X if there exists a sequence of stopping times (Tn )n>O such that o = TO ~ T 1 ~ T 2 ~ '" ~ Tn ~ .,. and limn->oo Tn = 00 a~., and such that X Tn l{Tn>o} has property 1f for each n. Since we are assuming our semi- martingales X satisfy Xo = 0, we could as well require only that X Tn has property 1f for each n. A related condition is that a property hold prelocally. Definition. A property 1f is said to hold prelocally for a process X with Xo = 0 if there exists a sequence of stopping times (Tn)n>l increasing to 00 a.s. such that X Tn - has property 1f for each n 2: 1. - Recall that XT- = X t l{o::;t 0: [M,M]t > n or it IdAsl > n} and let Y = X Tn -. Then Y has bounded jumps and hence it is a special semimartingale (Theorem 31 of Chap. III). Moreover or Y=L+C, where L = M Tn and C = A Tn - - (~MTn )I[Tn,oo)' Then [L, L] ~ n + (32, so L is a martingale in H 2 (Corollary 4 to Theorem 27 of Chap. II), and also hence C E H 2 by the lemma. Therefore X Tn - = L + C E H 2 . o 2 Stochastic Integration for Predictable Integrands 163 We are now in a position to define the stochastic integral for an arbi- trary semimartingale, as well as for predictable processes which need not be bounded. Let X be a semimartingale in 1t2 . To define a stochastic integral for pre- dictable processes H which are not necessarily bounded (written H E P), we approximate them with Hn E bP. Definition. Let X E H 2 with canonical decomposition X = N +A. We say H E P is (1t2 , X) integrable if Theorem 14. Let X be a semimartingale and let H E P be (H2, X) inte- grable. Let Hn = Hl{IHI:c;n} E bP. Then Hn . X is a Cauchy sequence in H 2 . Proof. Since Hn E bP, each n, the stochastic integrals Hn . X are defined. Note also that limn-+ooHn = H and that IHnl ~ IHI, each n. Then IIHn.X - H m . XII7-l2 = dx(Hn,Hm ) = 11(100 (H'; - H:,)2d[N,N]s)1/21IL2 + 11100 IH'; - H:'lldAs III L 2, and the result follows by two applications of the Dominated Convergence Theorem. 0 Definition. Let X be a semimartingale in 1t2 , and let H E P be (1t 2 , X) integrable. The stochastic integral H . X is defined to be limn -+oo Hn . X, with convergence in H2, where Hn = Hl{IH!:C;n}' Note that H . X in the preceding definition exists by Theorem 14. We can "localize" the above theorem by allowing both more general H E P and arbitrary semimartingales with the next definition. Definition. Let X be a semimartingale and H E P. The stochastic integral H .X is said to exist if there exists a sequence of stopping times Tn increasing to 00 a.s. such that X Tn - E H 2 , each n ~ 1, and such that H is (H 2,XTn -) integrable for each n. In this case we say H is X integrable, written H E L(X), and we define the stochastic integral by each n. Note that if m > n then 164 IV General Stochastic Integration and Local Times where Hk = Hl{IHI~k}, by the corollary of Theorem 8. Hence taking limits we have H· (XTTn-fn- = H· (XTn _), and the stochastic integral is well-defined for H E L(X). Moreover let R£ be another sequence of stopping times such that X R£ - E 1t2 and such that H is (1t2 , X R£ -) integrable, for each e. Again using the corollary of Theorem 8 combined with taking limits we see that on [0, R£ 1\ Tn), each e~ 1 and n ~ 1. Thus in this sense the definition of the stochastic integral does not depend on the particular sequence of stopping times. If H E bP (Le., H is bounded), then H E L(X) for all semimartingales X, since every semimartingale is prelocally in H 2 by Theorem 13. Definition. A process H is said to be locally bounded if there exists a sequence of stopping times (sm)m~l increasing to 00 a.s. such that for each m ~ 1, (Ht!\sTnl{sTn>o})t~O is bounded. Note that any process in L is locally bounded. The next example is suffi- ciently important that we state it as a theorem. Theorem 15. Let X be a semimartingale and let HE P be locally bounded. Then H E L(X). That is, the stochastic integral H . X exists. Proof Let (sm)m>l' (Tn)n>l be two sequences of stopping times, each in- creasing to 00 a.S:, such that H STn l{sTn>o} is bounded for each m, and X Tn - E H2 for each n. Define R n = min(Sn, Tn). Then H = H Rn l{Rn>o} on (O,Rn ) and hence it is bounded there. Since XRn - charges only (O,Rn ), we have that H is (H2, X R n -) integrable for each n ~ 1. Therefore using the sequence Rn which increases to 00 a.s., we are done. 0 We now turn our attention to the properties of this more general integral. Many of the properties are simple extensions of earlier theorems and we omit their proofs. Note that trivially the stochastic integral H . X, for H E L(X), is also a semimartingale. Theorem 16. Let X be a semimartingale and let H, J E L(X). Then o:H + (3J E L(X) and (o:H + (3J) . X = o:H· X + (3J. X. That is, L(X) is a linear space. Proof Let (Rm) and (Tn) be sequences of stopping times such that H is (H 2 , X R Tn -) integrable, each m, and J is (H 2 , X Tn -) integrable, each n. Tak- ing sn = Rn 1\ Tn, it is easy to cheek that o:H + (3J is (1t2 , X sn -) integrable for each n. 0 Theorem 17. Let X, Y be semimartingales and suppose H E L(X) and HE L(Y). Then H E L(X + Y) and H· (X + Y) = H· X + H . Y. 2 Stochastic Integration for Predictable Integrands 165 Theorem 18. Let X be a semimartingale and H E L(X). The jump process (tl(H . X)s)s~o is indistinguishable from (Hs(tlXs))s~o. Theorem 19. Let T be a stopping time, X a semimartingale, and HE L(X). Then (H· xf = Hl[o,T] . X = H . (XT ). Moreover, letting 00- equal 00, we have moreover Theorem 20.. Let X be a semimartingale with paths of finite variation on compacts. Let HE L(X) be such that the Stieltjes integral J~ IHslldXsl exists a.s., each t ?: O. Then the stochastic integral H . X agrees with a path-by-path Stieltjes integral. Theorem 21 (Associativity). Let X be a semimartingale with K E L(X). Then H E L(K· X) if and only if HK E L(X), in which case H· (K· X) = (HK) ·X. Theorem 22. Let X, Y be semimartingales and let H E L(X), K E L(Y). Then [H. X, K . Y]t = it HsKsd[X, Y]s (t 2: 0). Note that in Theorem 22 since H . X and H . Y are semimartingales, the quadratic covariation exists and the content of the theorem is the formula. Indeed, Theorem 22 gives a necessary condition for H to be in L(X), namely that J~ H;d[X, X]s exists and is finite for all t ?: O. The next theorem (The- orem 23) is a special case of Theorem 25, but we include it because of the simplicity of its proof. Theorem 23. Let X be a semimartingale, let H E L(X), and suppose Q is another probability with Q ~ P. If HQ . X exists, it is Q indistinguishable from Hp' X. Proof H Q . X denotes the stochastic integral computed under Q. By Theo- rem 14 of Chap. II, we know that H Q . X = H p ' X for H E L, and therefore if X E H 2 for both P and Q, they are equal for H E bP by the corollary of Theorem 5. Let (Rf )f>l, (Tn)n>l be two sequences of stopping times in- creasing to 00 a.s. such that H is (H2 , X R i -) integrable under Q, and H is (H 2 , X Tn -) integrable under P, each £ and n. Let 8 m = R m 1\ T m , so that H is (H 2 , Xs=-) integrable under both P and Q. Then H . X = limn->oo Hn . X on [0,8m ) in both dx(P) and dx(Q), where Hn = Hl{IHI::;n} E bP. Since Hp .X = H'Q . X, each n, the limits are also equal. 0 Much more than Theorem 23 is indeed true, as we will see in Theorem 25, which contains Theorem 23 as a special case. We need several preliminary results. 166 IV General Stochastic Integration and Local Times Lemma. Let X E H 2 and X = N + A be its canonical decomposition. Then Proof First observe that [X, X] = [N,N] + 2[N, A] + [A,A]. It suffices to show E{[N, A]oe} = 0, since then E{[N, N]oe} = E{[X, X]oe - [A, A]oe}, and the result follows since [A, A]oe ~ o. Note that by the Kunita-Watanabe inequalities. Also E{[M,A]oe} = 0 for all bounded martingales because A is natural. Since bounded martingales are dense in the space of L 2 martingales, there exists a sequence (Mn )n~l of bounded martingales such that limn->oe E{[Mn - N, Mn - N]oe} = o. Again using the Kunita-Watanabe inequalities we have E{I[N - Mn,A]oel} ~ (E{[N - Mn,N - M n]oe})1/2(E{[A,A]oe})1/2 and therefore limn->oe E{[Mn, A]oe} = E{[N, A]oe}. Since E{[Mn, A]oe} = 0, each n, it follows that E{[N, Aloe} = o. 0 Note that in the preceding proof we established the useful equality E{[X, Xlt} = E{[N, N]t} + E{[A, A]t} for a semimartingale X E 1t2 with canonical decomposition X = N + A. Theorem 24. For X a semimartingale in H 2 , Proof By Theorem 5 for jHI ~ 1 Since 211[X,X]~21IL2 ~ 211[M,M]~2I1L2 + 21110e IdAs lll£2 = 21IXII7-t2 , where X = M + A is the canonical decomposition of X, we have the right inequality. For the left inequality we have II[M,Ml~21IL2 ~ II[X,X]~21IL2 by the lemma preceding this theorem. Moreover if IHI ~ 1, then 2 Stochastic Integration for Predictable Integrands 167 II(H . AML2 ~ II(H . XM£2 + II(H . MML2 ~ II(H· X)~II£2 + II[M,M]~211£2' Next take H = ~; this exists as a predictable process since A is predictable, and therefore A predictable implies that IAI, the total variation process, is also predictable. Consequently we have IIXII?-t2 = II[M,M]~211£2 + 11100 IdAs III L2 = II[M,M]~2I1L2 + II(H· A)ooll£2 ~ II[M,M]~211£2 + II(H. X)~II£2 + II[M,M]~211£2 ~ II(H. X)~IIL2 + 211[X,X]~21IL2. 0 We present as a corollary to Theorem 24, the equivalence of the two pseudonorms SUPIHI911(H . X)~IIL2 and IIXII?-t2. We will not have need of this corollary in this book,5 but it is a pretty and useful result nevertheless. It is originally due to Yor [241], and it is actually true for all p, 1 ~ P < 00. (See also Dellacherie-Meyer [46, pages 303-305].) Corollary. For a semimartingale X (with Xo = 0), and in particular SUPIHI9 II(H . X)~II£2 < 00 if and only if IIXII?-t2 < 00. Proof. By Theorem 5 if IIXII?-t2 < 00 and IHI ~ 1 we have Thus we need to show only the left inequality. By Theorem 24 it will suffice to show that II[X,X]~21IL2 ~ sup II(H ,X)~II£2, IHI9 for a semimartingale X with Xo = O. To this end fix at> a and let a = To ~ T1 ~ ... ~ Tn = t be a random partition of [0, t]. Choose Eb ... ,En non-random and each equal to 1 or -1 and let H = L:~=l Ei 1(T;-l ,T;]' Then H is a simple predictable process and n (H· X)oo = L Ei(XTi - X Ti _ 1 )· i=l Let 0: = SUPIHI911(H· X)~II£2' We then have 5 However the corollary does give some insight into the relationship between The- orems 12 and 14 in Chap. V. 168 IV General Stochastic Integration and Local Times n 0:2 ::::: E{(H. X)~} = 2:.::: EiEjE{(XTi - XTi_l)(XTj - X Tj _1)}· i,j=l If we next average over all sequences Cl, ... ,Cn taking values in the space {±l}n, we deduce n n 0:2 ::::: 2:.::: E{ (XTi - X Ti _1)2} = E{2:.:::(XTi - X Ti _J2}. i=l i=l Next let CTm = {Tim} be a sequence of random partitions of [O,t] tending to the identity. Then Li(XTt, -XTi':':1)2 converges in probability to [X,XJt, Let {md be a subsequence so that Li(XT"'k - XT,!,k)2 converges to [X, X]t a.s. 1. 1.-1 Finally by Fatou's Lemma we have E{[X, X]tl ::; liminf E{'""'(XT"'k - XT,!,k )2} ::; 0:2. mk---'7(X) L...t t 1.-1 i Letting t tend to 00 we conclude that E{[X,X]oo}::; 0:2 = sup II(H· X)~lli2. IHI9 It remains to show that if SUPIHI911(H. X)~II£2 < 00, then X E le. We will show the contrapositive. If X f/- 1{2, then SUPIHISlll(H . X)~II£2 = 00. Indeed, let Tn be stopping times increasing to 00 such that X Tn - is in 1{2 for each n (cf., Theorem 13). Then IIXTn-II?-t2 ::; 3 sup II(H· XTn_)~II£2 ::; 91IXTn-II?-t2. IHI9 Letting n tend to 00 gives the result. o Before proving Theorem 25 we need two technical lemmas. Lemma 1. Let A be a non-negative increasing FV process and let Z be a positive uniformly integrable martingale. Let T be a stopping time such that A = AT - (that is, Aoo = AT -) and let k be a constant such that Z ::; k on [0, T). Then Proof. Since Ao- = Zo- = 0, by integration by parts 2 Stochastic Integration for Predictable Integrands 169 where the second integral in the preceding is a path-by-path Stieltjes integral. Let Rn be stopping times increasing to 00 a.s. that reduce the local martingale U~ As_dZsk:~o· Since dAs charges only [0, T) we have it ZsdAs ::::; it kdAs ::::; kAoo for every t ::::: O. Therefore E{(AZ)Rn} = E{l Rn As_dZs} + E{l Rn ZsdAs} ::::; 0+ E{kAoo }. The result follows by Fatou's Lemma. o Lemma 2. Let X be a semimartingale with Xo = 0, let Q be another prob- ability with Q ~ P, and let Zt = Ep{~IFt}. If T is a stopping time such that Zt ::::; k on [0, T) for a constant k, then Proof By Theorem 24 we have II XT-II?t2 (Q)::::; sup EQ{((H.XT-)*)2}1 +2EQ{[XT-,XT-j4} IHl9 1 ::::; sup Ep{dQ((H'XT-)*)2}2+2Ep{dQ[XT-,XT-]~} IHI~l dP dP ::::; Vk sup Ep{((H.XT-)*)2}~+2VkEp{[XT-,XT-]~}. IHI9 where we have used Lemma 1 on both terms to obtain the last inequality above. The result follows by the right inequality of Theorem 24. 0 Note in particular that an important consequence of Lemma 2 is that if Q « P with !!fj; bounded, then X E 1f2(p) implies that X E 1f2(Q) as well, with the estimate IIXII?t2(Q) ::::; 5JkIIXII?t2(p), where k is the bound for ~~. Note further that this result (without the estimate) is obvious if one uses the equivalent pseudonorm given by the corollary to Theorem 24, since sup EQ{((H. X)~Y} = sup Ep{dQp((H. X)~)2} IHI9 IHI9 d ::::; k sup Ep{((H· X)~)2}, IHI~l where again k is the bound for ~~. 170 IV General Stochastic Integration and Local Times Theorem 25. Let X be a semimartingale and H E L(X). If Q ~ P, then HE L(X) under Q as well, and HQ . X = H p . X, Q-a.s. Proof Let Tn be a sequence of stopping times increasing to 00 a.s. such that His (XTn _, Jt2) integrable under P, each n 2: 1. Let Zt = Ep{~IFtl, the cadlag version. Define sn = inf{t > 0 : IZt I > n} and set Rn = sn /\ Tn. Then X Rn - E 1i2(P) n1i2(Q) by Lemma 2, and H is (1i2,XRn -) integrable under P. We need to show H is (1i2 ,XRn -) integrable under Q, which will in turn imply that H E L(X) under Q. Let X Rn - = N + C be the canonical decomposition under Q. Let Hm = Hl{IHI$m}' Then (EQ{!(Hr;')2d[N,N]s})1/2 + II! IHr;'lldCslll£2(Q) m R U r::: m Rn = IIH .X -11?t2(Q)::; 5ynliH .X -1I?t2(p) r::; R n ::; 5y nliH . X -1I?t2(p) < 00, and then by monotone convergence we see that H is (1i2 , X R n -) integrable under Q. Thus H E L(X) under Q, and it follows that HQ . X = H p . X, Q-a.s. 0 Theorem 25 can be used to extend Theorem 20 in a way analogous to the extension of Theorem 17 by Theorem 18 in Chap. II. Theorem 26. Let X, X be two semimartingales, and let H E L(X), H E L(X). Let A = {w : H.(w) = H.(w) and x.(w) = X.(w)}, and let B = {w : t 1--4 Xt(w) is of finite variation on compacts}. Then H·X = H·X on A, and H . X is equal to a path-by-path Lebesgue-Stieltjes integral on B. Proof Without loss of generality assume P(A) > O. Define Q by Q(A) = P(AIA). Then Q ~ P and therefore H E L(X), H E L(X) under Q as well as under P by Theorem 25. However under Q the processes Hand H as well as X and X are indistinguishable. Thus HQ'X = HQ·X and hence J{·X = H·X P-a.s. on A by Theorem 25, since Q ~ P. The second assertion has an analogous proof (see the proof of Theorem 18 of Chap. II). 0 Note that one can use stopping times to localize the result of Theorem 26. The proof of the following corollary is analogous to the proof of the corollary of Theorem 18 of Chap. II. Corollary. With the notation of Theorem 26, let S, T be two stopping times with S < T. Define C = {w : Ht(w) = Ht(w); Xt(w) = Xt(w); S(w) < t ::; T(w)}, D = {w : t 1--4 Xt(w) is of finite variation on S(w) < t < T(w)}. Then H· XT - H· XS = H· X T - H· X S on C and H· X T - H· X S equals a path-by-path Lebesgue-Stieltjes integral on D. 2 Stochastic Integration for Predictable Integrands 171 Theorem 27. Let Pk be a sequence of probabilities such that X is a Pk semimartingale for each k. Let R = 2::~1 AkPk where Ak ::::: 0, each k, and 2::%':1 Ak = 1. Let H E L(X) under R. Then H E L(X) under Pk and HR' X = H pk . X, Pk-a.s., for all k such that Ak > O. Proof. If Ak > 0 then Pk « R. Moreover since Pk(A) ::; }k R(A), it follows that H E L(X) under Pk . The result then follows by Theorem 25. 0 We now turn to the relationship of stochastic integration to martingales and local martingales. In Theorem 11 we saw that if M is a square integrable martingale and H E bP, then H· M is also a square integrable martingale. When M is locally square integrable we have a simple sufficient condition for H to be in L(M). Lemma. Let M be a square integrable martingale and let H E P be such that E{fooo H;d[M, M]s} < 00. Then H· M is a square integrable martingale. Proof. If H k E bP, then Hk . M is a square integrable martingale by The- orem 11. Taking Hk = Hl{IHI$k}' and since H is (1{2, M) integrable, by Theorem 14 H k . M converges in 1{2 to H . M which is hence a square inte- grable martingale. 0 Theorem 28. Let M be a locally square integrable local martingale, and let H E P. The stochastic integral H . M exists (i. e., H E L(M)) and is a locally square integrable local martingale if there exists a sequence of stopping times (Tn)n~l increasing to 00 a.s. such that E{foTn H";d[M, M]s} < 00. Proof. We assume that M is a square integrable martingale stopped at the time Tn. The result follows by applying the lemma. 0 Theorem 29. Let M be a local martingale, and let HE P be locally bounded. Then the stochastic integral H . M is a local martingale. Proof. By stopping we may, as in the proof of Theorem 29 of Chap. III, assume that H is bounded, M is uniformly integrable, and that M = N + A where N is a bounded martingale and A is of integrable variation. We know that there exists R k increasing to 00 a.s. such that M Rk - E 1{2, and since HE bP there exist processes Hi E bJL such that IIHi. MRk- - H· MRk-II?t2 tends to zero. In particular, Hi . MRk- tends to H . MRk- in ucp. Therefore we can take Hi such that Hi . M Rk - tends to H· M in ucp, with Hi E bJL. Finally without loss of generality we assume Hi . M converges to H . M in ucp. Since Hi . M = Hi . N + Hi . A and Hi . N converges to H . N in ucp, we deduce Hi . A converges to H· A in ucp as well. Let 0 ::; s < t and assume Y E b.rs • Therefore, since A is of integrable total variation, and since Hi . A is a martingale for Hi E bJL (see Theorem 29 of Chap. III), we have 172 IV General Stochastic Integration and Local Times E{Y it HudAu} = E{Y it lim H~dAu} = E{Y lim it H~dAu} s+ s+ i-+oo i-+oo s+ = lim E{Y jt H~dAu} i-+oo s+ =0, where we have used Lebesgue's Dominated Convergence Theorem both for the Stieltjes integral w-by-w (taking a subsequence if necessary to have a.s. convergence) and for the expectation. We conclude that U~ HsdAs}t-?o is a martingale, hence H . M = H . N + H . A is also a martingale under the assumptions made; therefore it is a local martingale. 0 In the proof of Theorem 29 the hypothesis that H E P was locally bounded was used to imply that if M = N + A with N having locally bounded jumps and A an FV local martingale, then the two processes are locally integrable. Thus one could weaken the hypothesis that H is locally bounded, but it would lead to an awkward statement. The general result, that M a local martingale and HE L(M) implies that H· M is a local martingale, is not true! See Emery's example, which precedes Theorem 34 for a stochastic integral with respect to an H 2 martingale which is not even a local martingale! Emery's counterexample has lead to the development of what are now known as sigma martingales, a class of processes which are not local martingales, but can be thought to be "morally local martingales." This is treated in Section 9 of this chapter. Corollary. Let M be a local martingale, Mo = 0, and let T be a predictable stopping time. Then M T - is a local martingale. Proof. The notation M T- means Mr- = Mt1{t 2 Stochastic Integration for Predictable Integrands 173 Note that if M is a local martingale and T is an arbitrary stopping time, it is not true in general that M T - is still a local martingale. Since a continuous local martingale is locally square integrable, the theory is particularly nice. Theorem 30. Let M be a continuous local martingale and let H E P be such that I; H;d[M, M]s < 00 a.s., each t ~ O. Then the stochastic integral H . M exists (i.e., HE L(M)) and it is a continuous local martingale. Proof Since M is continuous, we can take Rk = inf{t > 0: IMtl > k}. Then IMti\Rk I ::; k and therefore M is locally bounded, hence locally square integrable. Also M continuous implies [M, M] is continuous, whence if T k = inf{t > 0: ht H;d[M,M]s > k}, we see that U; H;d[M, M]s k~o is also locally bounded. Then H· M is a locally square integrable local martingale by Theorem 28. The stochastic integral H . M is continuous because fl(H.M) = H(flM) and flM = 0 by hypothesis. 0 In the classical case where the continuous local martingale M equals B, a standard Brownian motion, Theorem 30 yields that if HE P and I~ H;ds < 00 a.s., each t ~ 0, then the stochastic integral (H . Btk~_o = (J~ HsdBs)t?o exists, since [B, B]t = t. Corollary. Let X be a continuous semimartingale with (unique) decomposi- tion X = M + A. Let H E P be such that ht H;d[M, M]s +ht IHslldAsl < 00 a.s. each t ~ O. Then the stochastic integral (H· X)t = I~ HsdXs exists and it is continuous. Proof. By the corollary of Theorem 31 of Chap. III, we know that M and A have continuous paths. The integral H· M exists by Theorem 30. Since H· A exists as a Stieltjes integral, it is easy to check that H E L(A), since A is continuous, and the result follows from Theorem 20. 0 In the preceding corollary the semimartingale X is continuous, hence [X, Xl = [M, M] and the hypothesis can be written equivalently as I t H;d[X, X]s +I t IHslldAsl < 00 a.s. each t ~ O. We end our treatment of martingales with a special case that yields a particularly simple condition for H to be in L(M). 174 IV General Stochastic Integration and Local Times Theorem 31. Let M be a local martingale with jumps bounded by a constant (3. Let HE P be such that I~ H;d[M,M]s < 00 a.s., t::::: 0, and E{Hf} < 00 for any bounded stopping time T. Then the stochastic integral (J~ HsdMsk~.o exists and it is a local martingale. Proof Let Rn = inf{t > a : I~ H;d[M, M]s > n}, and let Tn = min(Rn , n). Then rn are bounded stopping times increasing to 00 a.s. Note that Tn E{1 H;d[M, M]s} ::; n + E{Hfn (LlMTn )2} ::; n + (32 E{Hfn} < 00, and the result follows from Theorem 28. o The next theorem is, of course, an especially important theorem, the Dom- inated Convergence Theorem for stochastic integrals. Theorem 32 (Dominated Convergence Theorem). Let X be a semi- martingale, and let H m E P be a sequence converging a.s. to a limit H. If there exists a process G E L(X) such that IHm / ::; G, all m, then H m , H are in L(X) and H m . X converges to H· X in ucp. Proof. First note that if /JI ::; G with J E P, then J E L(X). Indeed, let (Tn)n>l increase to 00 a.s. such that G is (1i 2 ,XTn -) integrable for each n. Then clearly E{l°O J;d[N, N]s} + E{(l°O IJs lldAs l)2} ::; E{1°O G;d[N, N]s} + E{(1°O IGslldAsl)2} < 00, and thus J is (1i2 , X Tn -) integrable for each n. (Here N +A is the canonical decomposition of X Tn - .) To show convergence in ucp, it suffices to show uniform convergence in probability on intervals of the form [0, to] for to fixed. Let E: > abe given, and choose n such that P(Tn < to) < E:, where X Tn - E 1i2 and G is (1i 2 , X Tn -) integrable. Let X Tn - = N + A, the canonical decomposition. Then E{supIHm . X Tn - _ H. X Tn -1 2 } t~to The second term tends to zerO by Lebesgue's Dominated Convergence Theo- rem. Since IHm - HI ::; 2G, the integral (Hm - H) . N is a square integrable martingale (by the lemma preceding Theorem 28). Therefore using Doob's maximal quadratic inequality, we have 2 Stochastic Integration for Predictable Integrands 175 E{sup I(Hm - H)· NI 2}::; 4E{I(Hm - H). N to l 2 } t~to and again this tends to zero by the Dominated Convergence Theorem. Since convergence in £2 implies convergence in probability, we conclude for c5 > 0, limsupP{sup IHm . X t - H· Xtl > c5} m-HXJ t~to ::; lim sup P{sup 1((Hm - H) . XTn-)tl > c5} + P(Tn < to) m---+oo t~to and since E: is arbitrary, the limit is zero. o We use the Dominated Convergence Theorem to prove a seemingly in- nocuous result. Generalizations, however, are delicate as we indicate following the proof. Theorem 33. Let IF = (Ftk~o and G = (Qt)r~o be two filtrations satisfying the usual hypotheses and suppose F t C Qt, each t ::::: 0, and that X is a semimartingale for both IF and G. Let H be locally bounded and predictable for IF. Then the stochastic integrals H IF • X and H'G . X both exist, and they are equal. 6 Proof. It is trivial that H is locally bounded and predictable for (Qt}t>o as well. By stopping, we can assume without loss of generality that H is bounded. Let 1i = {all bounded, :F predictable H such that H IF • X = H'G . X}. Then 1i is clearly a mOnotone vector space, and 1i contains the multiplicative class bJL by Theorem 16 of Chap. II. Thus using Theorem 32 and the Monotone Class Theorem we are done. 0 It is surprising that the assumption that H be locally bounded is impor- tant. Indeed, Jeulin [114, pages 46,47] has exhibited an example which shows that Theorem 33 is false in general. Theorem 33 is not an exact generalization of Theorem 16 of Chap. II. Indeed, suppose IF and G are two arbitrary filtrations such that X is a semi- martingale for both IF and G, and H is bounded and predictable for both of them. If It = Ft n Qt, then X is still an (It)t~o semimartingale by Stricker's Theorem, but it is not true in general that H is (It)t~o predictable. It is an open question as to whether or not HF . X = Hg . X in this situation. For a partial result, see Zheng [248]. 6 HF • X and HG . X denote the stochastic integrals computed with the filtrations (.Ft)t~O and (gt)t~O' respectively. 176 IV General Stochastic Integration and Local Times Example (Emery's example of a stochastic integral behaving badly). The following simple example is due to M. Emery, and it has given rise to the study of sigma martingales, whose need in mathematical finance has become apparent. Let X = (Xtk~o be a stochastic process given by the following description. Let T be an exponential random variable with parameter A = 1, let U be an independent random variable such that P {U = I} = P {U = -I} = 1/2, and set X = U1 {t~T}' Then X together with its minimal filtration satisfying the usual hypotheses is a martingale in H2 . That is, X is a stopped compound Poisson process with mean zero and is an L2 martingale. Let H t = t1{t>O}' Therefore H is a deterministic integrand, continuous on (0,00), and hence predictable. Consequently the path-by-path Lebesgue-Stieltjes integral Zt = I~ HsdXs exists a.s. However H . X is not locally in HP for any p 2: 1. (However since it is still a semimartingale7 , it is prelocally in H2 .) Moreover, even though X is an L 2 martingale and H is a predictable integrand, the stochastic integral H . X is not a local martingale because E{IZsl} = 00 for every stopping time S such that P(S > 0) > O. The next theorem is useful, since it allows one to work in the convenient space H2 through a change of meaSUre. It is due originally to Bichteler and Dellacherie, and the proof here is due to Lenglart. We remark that the use of the exponent 2 is not important, and that the theorem is true for HP for any p2:1. Theorem 34. Let X be a semimartingale on a jiltered complete probability space (D, F, IF, P) satisfying the usual hypotheses. Then there exists a proba- bility Q which is equivalent to P such that under Q, X is a semimartingale in H 2 . Moreover, !f!i; can be taken to be bounded. Before we begin the proof we establish a useful lemma. Lemma. Let (D, F, P) be a complete probability space and let X n be a se- quence of a.s. finite valued random variables. There exists another probability Q, equivalent to P and with a bounded density, such that every X n is in L 2 (dQ). Proof. Assume without loss that each of the X n is positive. For a single ran- dom variable X we take Ak = {k ::; X < k + I} and Y = Lk>l T k 1Ak' Then Y is bounded and - EQ{X2} = E{X2y}::; 2::: (k ;k1)2 < 00. k~l For the general case, choose constants an such that P(Xn > an) ::; 2-n. The Borel-Cantelli Lemma implies that a.s. X n ::; an for all n sufficiently large. Next choose constants en such that Ln~l cnan < 00. Let Yn be the bounded 7 H. X is still a semimartingale since it is of finite variation a.s. on compact time sets. 2 Stochastic Integration for Predictable Integrands 177 density chosen for X n individually as done in the first part of the proof, and take Y = En2:1 cnYn so that we have the result. 0 Proof of Theorem 34. Using the lemma we make a first change of measure making all of the random variables [X, X]n integrable. Recall that [X, X] is invariant under a change to an equivalent probability measure. By abusing notation, we will still denote this new measure by P. This implies that if Jt = sUPs 178 IV General Stochastic Integration and Local Times 3 Martingale Representation In this section we will be concerned with martingales, rather than semimartin- gales. The question of martingale representation is the following. Given a col- lection A of martingales (or local martingales), when can all martingales (or all local martingales) be represented as stochastic integrals with respect to processes in A? This question is surprisingly important in applications, and it is particularly interesting (in finance theory, for example) when A consists of just one element. Throughout this section we assume as given an underlying complete, fil- tered probability space (fl, F, (Ft k::°,P) satisfying the usual hypotheses. We begin by considering only L 2 martingales. Later we indicate how to extend these results to locally square integrable local martingales. Definition. The space M 2 of L 2 martingales is all martingales M such that SUPt E {M[} < 00, and Mo = 0 a.s. Notice that if M E M 2 , then limhooE{M[} = E{M~} < 00, and Mt = E{MooIFtl. Thus each M E M 2 can be identified with its terminal value Moo. We can endow M 2 with a norm and also with an inner product for M, N E M 2 . It is evident that M 2 is a Hilbert space and that its dual space is also M 2 . If E{M[} < 00 for each t, we call M a square integrable martingale. If in addition SUPt E{M[} = E{M~} < 00, then we call Man L 2 martingale. The next definition is a key idea in the theory of martingale representation. It differs slightly from the customary definition because we are assuming all martingales are zero at time t = O. If we did not have this hypothesis, we would have to add the condition that for any event A E Fo, any martingale M in the subspace F, then MIA E F. Definition. A closed subspace F of M 2 is called a stable subspace if it is stable under stopping (that is, if M E F and if T is a stopping time, then M T E F).8 Theorem 35. Let F be a closed subspace of M 2 . Then the following are equivalent. (a) F is closed under the following operation. For M E F, (M - M t )lA E F for A E Ft> any t ~ O. (b) F is a stable subspace. 8 Recall that MT = MtI\T, and Mo = O. 3 Martingale Representation 179 (c) If ME F and H is bounded, predictable, then (f~ HsdMsh-z.o = H . ME F. (d) If M E F and H is predictable with E{fooo H;d[M, M]s} < 00, then H·MEF. Proof. Property (d) implies (c), and it is simple that (c) implies (b). To get (b) implies (a), let T = tA, where { t, tA = 00, if wE A, if w tJ- A. Then T = tA is a stopping time when A EFt, and (M - M t )lA = M - M T ; since F is assumed stable, both M and MT are in F. It remains to show only that (a) implies (d). Note that if H is simple predictable of the special form n H t = 1Ao1{o} + L 1Ai 1(t"ti+tl i=l with Ai E Fti , 0 ::; i ::; n, 0 = to ::; tl ::; '" ::; tn+l < 00, then H . M E F whenever M E F. Linear combinations of such processes are dense in bJL which in turn is dense in bP under dM (·,·) by Theorem 2. But then bP is dense in the space of predictable processes 1{ such that E{fooo H;d[M, M]s} < 00, as is easily seen (d., Theorem 14). Therefore (a) implies (d) and the theorem is proved. 0 Since the arbitrary intersection of closed, stable subspaces is still closed and stable, we can make the following definition. Definition. Let A be a subset of M 2. The stable subspace generated by A, denoted S(A), is the intersection of all closed, stable subspaces containing A. As already noted on page 178, we Can identify a martingale M E M 2 with its terminal value Moo E L 2 . Therefore another martingale N E M 2 is (weakly) orthogonal to M if E{NooMoo } = O. There is however another, stronger notion of orthogonality for martingales in M 2 . Definition. Two martingales N, M E M 2 are said to be strongly orthog- onal if their product L = NM is a (uniformly integrable) martingale. Note that if N, M E M 2 are strongly orthogonal, then N M being a (uniformly integrable) martingale implies that [N, M] is also a local martin- gale by Corollary 2 of Theorem 27 of Chap. II. It is a uniformly integrable martingale by the Kunita-Watanabe inequality (Theorem 25 of Chap. II). Thus M, N E M 2 are strongly orthogonal if and only if [M, N] is a uni- formly integrable martingale. If Nand M are strongly orthogonal then E{NooMoo } = E{Loo } = E{Lo} = 0, so strong orthogonality implies or- thogonality. The converse is not true however. For example let ME M 2 , and 180 IV General Stochastic Integration and Local Times let Y E Fo, independent of M, with P(Y = 1) = P(Y = -1) Nt = YMt , t ~ O. Then N E M 2 and E{NooMoo } = E{YM~J = E{Y}E{M';} = 0, ~. Let so M and N are orthogonal. However M N = YM 2 is not a martingale (unless M = 0) because E{YM;IFo} = YE{M;IFo} =f- 0 = YM6. Definition. For a subset A of M 2 we let Ai- (resp. AX) denote the set of all elements of M2 orthogonal (resp. strongly orthogonal) to each element of A. Lemma 1. If A is any subset ofM2, then AX is (closed and) stable. Proof Let M n be a sequence of elements of AX converging to M, and let N E A. Then M n N is a martingale for each n and A x will be shown to be closed if M N is also one, or equivalently that [M, N] is a martingale. However E{I[Mn,N] - [M,N]tl} = E{I[Mn -M,N]tl} ~ (E{[Mn - M, M n - M]t} )1/2(E{[N, N]t} )1/2 by the Kunita-Watanabe inequalities. It follows that [Mn, N]t converges to [M, N]t in Ll, and therefore [M, N] is a martingale, and A x is closed. Also A x is stable because MEA x, N E A implies [MT , N] = [M, NjT is a martingale and thus M T is strongly orthogonal to N. 0 Lemma 2. Let N, M be in M2. Then the following are equivalent. (a) M and N are strongly orthogonal. (b) S(M) and N are strongly orthogonal. (c) S(M) and S(N) are strongly orthogonal. (d) S(M) and N are weakly orthogonal. (e) S(M) and S(N) are weakly orthogonal. Proof If M and N are strongly orthogonal, let A = {N} and then MEA x . Since AX is a closed stable subspace by Lemma 1, S(M) c {NY. Therefore (b) holds and hence (a) implies (b). The same argument yields that (b) implies (c). That (c) implies (e) which implies (d) is obvious. It remains to show that (d) implies (a). Suppose N is weakly orthogonal to S(M). It suffices to show that [N, M] is a martingale. By Theorem 21 of Chap. I it suffices to show E{[N, M]T} = 0 for any stopping T. However E{[N, M]T} = E{[N, MT]oo} = 0, since N is orthogonal to M T which is in S(M). 0 Theorem 36. Let Ml, ... ,Mn E M 2 , and suppose M i , Mj are strongly or- thogonal for i =f- j. Then S(M1,oo.,Mn) consists of the set of stochastic integrals n HI . M 1+ ... + H n . M n = 2: Hi . M i , i=1 3 Martingale Representation 181 where Hi is predictable and 1:::; i :::; n. Proof. Let I denote the space of processes L~=IHi . Mi, where Hi satisfy the hypotheses of the theorem. By Theorem 35 any closed, stable subspace must contain I. It is simple to check that I is stable, so we need to show only that I is closed. Let Then the mapping (HI, H 2 , ... ,Hn ) -t L~=I Hi . Mi is an isometry from Llrl EEl ... EEl Llrn into M 2 . Since it is a Hilbert space isometry its image I is complete, and therefore closed. 0 Theorem 37. Let A be a subset of M 2 which is stable. Then AJ.. is a stable subspace, and if ME AJ.. then M is strongly orthogonal to A. That is, AJ.. = AX, and S(A) = AJ..J.. = AXJ.. = A XX . Proof We first show that AJ.. = AX. Let MEA and N E AJ... Since N is or- thogonal to S(M), by Lemma 2, Nand M are strongly orthogonal. Therefore AJ.. c AX. However clearly AX C AJ.., whence AX = AJ.., and thus AJ.. = AX is a stable subspace by Lemma 1. By the above applied to AJ.., we have that (AJ..)J.. = (AJ..)x. It remains to show that S(A) = AJ..J... Since AJ..J.. = A, the closure of A in M 2, it suffices to show that A is stable. However it is simple to check that condition (a) of Theorem 35 is satisfied for A, since it already is satisfied for A, and we conclude A is a stable subspace. 0 Corollary 1. Let A be a stable subspace of M2. Then each M E M 2 has a unique decomposition M = A + B, with A E A and B E A X • Proof A is a closed subspace of M 2, so each M E M2 has a unique decom- position into M = A + B with A E A and B E AJ... However AJ.. = AX by Theorem 37. 0 Corollary 2. Let M, N E M 2, and let L be the projection of N onto S(M), the stable subspace generated by M. Then there exists a predictable process H such that L = H . M. Proof We know that such an L exists by Corollary 1. Since {M} consists of just one element we can apply Theorem 36 to obtain the result. 0 182 IV General Stochastic Integration and Local Times Definition. Let A be finite set of martingales in M 2. We say that A has the (predictable) representation property if I = M 2 , where n I = {X : X = L Hi . M i, M i E A}, i=l each Hi predictable such that Corollary 3. Let A = {Ml, ... ,M n } C M 2 , and suppose M i , Mj are strongly orthogonal for i =f- j. Suppose further that if N E M 2 , N ..1 A in the strong sense implies that N = o. Then A has the predictable representation property. Proof. By Theorem 36 we have S(A) = I. The hypotheses imply that S(A)J.. = {O}, hence S(A) = M 2 . 0 Stable subspaces and predictable representation can be considered from an alternative perspective. Up to this point we have assumed as given and fixed an underlying space (fl, F, (Ft)t>o, P), and a set of martingales A in M 2 . We will see that the property th-;;;'t S(A) = M 2 , intimately related to predictable representation (cf., Theorem 36), is actually a property of the probability measure P, considered as one element among the collection of probability measures that make L 2 (Fth>o martingales of all the elements of A. Our first observation is that since the filtration IF = (Ftk~o is assumed to be P complete, it is reasonable to consider only probability measures that are absolutely continuous with respect to P. Definition. Let A C M2. The set of M2 martingale measures for A, denoted M 2 (A), is the set of all probability measures Q defined on Vo 3 Martingale Representation 183 Proof Let Q and R E M 2 (A), and let 8 = >.Q + (1 - >')R, 0 < >. < 1. Then for X E M 2 (A), sup Es{M;} = sup [>.EQ{M;} +(1 - >')ER{M;}] < 00, t t since Q, R E M 2 (A). Also if HE b.rs , s < t, then Es{MtH} = >.EQ{MtH} + (1 - >')ER{MtH} = >.EQ{MsH} + (1 - >')ER{MsH} = Es{MsH}, o Definition. A measure Q E M 2 (A) is an extremal point of M 2 (A) if whenever Q = >.R + (1 - >.)8 with R, 8 E M 2 (A), R -=I- 8, 0 ~ >. ~ 1, then >. = 0 or 1. Theorem 38. Let A C M2. If S(A) = M 2 then P is an extremal point of M 2 (A). Proof. Suppose P is not extremal. We will show that S(A) -=I- M 2 . Since P is not extremal, there exist Q, R E M 2 (A), Q -=I- R, such that P = >'Q+(l->')R, 0< >. < 1. Let dQ L oo = dP' and let Lt = E{~ l.rd. Then 1 = ~~ = >.Loo + (1- >.) ~~ ~ >.Loo a.s., so that L oo ~ t a.s. Therefore L is a bounded martingale with Lo = 1 (since Q = P on .ro), and thus L - Lo is a nonconstant martingale in M 2 (P). However, if X E A and HE b.rs , then X is a Q martingale and for s < t, Ep{XtLtH} = Ep{XtLooH} = EQ{XtH} = EQ{XsH} = Ep{XsLooH} = Ep{XsLsH}, and XL is a P martingale. Therefore X(L - L o) is a P martingale, and L - Lo E M 2 and it is strongly orthogonal to A. By Theorem 37 we cannot have S(A) = M 2 . 0 Theorem 39. Let A C M 2 . If P is an extremal point of M 2 (A), then every bounded P martingale strongly orthogonal to A is null. Proof. Let L be a bounded nonconstant martingale strongly orthogonal to A. Let c be a bound for ILl, and set ( LoodQ= l--)dP2c and dR = (1 + L oo )dP. 2c We have Q, R E M 2 (A), and P = ~Q + ~R is a decomposition that shows that P is not extremal which is a contradiction. 0 184 IV General Stochastic Integration and Local Times Theorem 40. Let A = {M1 , ... , M n } C M2, with Mi continuous and Mi, Mj strongly orthogonal for i =f- j. Suppose P is an extremal point of M 2 (A). Then (a) every stopping time is accessible; (b) every bounded martingale is continuous; (c) every uniformly integrable martingale is continuous; and (d) A has the predictable representation property. Proof. (a) Suppose T is a totally inaccessible stopping time and P(T < 00) > o. By Theorem 22 of Chap. III, there exists a martingale M with tlMT = l{T 0: IMtl > n}, 1MR n I ::; n + 1 and thus M is locally bounded. Indeed, we then have M R n are bounded martingales strongly orthogonal to Mi. By Theorem 39 we have MR n = 0 for each n. Since limn ---.oo Rn = 00 a.s., we conclude M = 0, a contradiction. (b) Let M be a bounded martingale which is not continuous, and assume Mo = O. Let Te = inf{t > 0 : ItlMtl > c}. Then there exists c > 0 such that for T = Te, P{ltlMTI > O} > O. By part (a) the stopping time T is accessible, hence without loss we may assume that T is predictable. Therefore M T - is a bounded martingale by the corollary to Theorem 29, whence N = M T - M T- = tlMT l{tzT} is also a bounded martingale. However N is a finite variation bounded martingale, hence [N, Mi] = 0 each i. That is, N is a bounded martingale strongly orthogonal to A. Hence N = M T - M T - = 0 by Theorem 39, and we conclude that M is continuous. (c) Let M be a uniformly integrable martingale closed by Moo. Define Then Mn are bounded martingales and therefore continuous by part (b). However 1 P{sup 1M;' - Mtl > c} ::; - E{IM~ - Mool} t c by an inequality of Doob10 , and the right side tends to 0 as n tends to 00. Therefore there exists a subsequence (nk) such that limk---.oo M;'k = M t a.s., uniformly in t. Thus M is continuous. (d) By Corollary 3 of Theorem 37 it suffices to show that if N E A x then N = O. Suppose N E A x. Then N is continuous by (c). Therefore N is locally bounded, and hence by stopping, N must be 0 by Theorem 39. 0 10 See, for example, Breiman [23, page 88]. 3 Martingale Representation 185 The next theorem allows us to consider subspaces generated by countably infinite collections of martingales. Theorem 41. Let M E M 2, yn E M 2, n 2:: 1, and suppose y~ converges to Yoo in L2, and that there exists a sequence Hn E L(M) such that ~n = I~ Hr;dMs' n 2:: 1. Then there exists a predictable process H E L(M) such that yt = I; HsdMs. Proof If Y~ converges to Y00 in L2, then y n converges to Y in M 2 • By Theorem 36 we have that S(M) = I(M), the stochastic integrals with respect to M. Moreover yn E S(M), each n. Therefore Y is in the closure of S(M); but S(M) is closed, so Y E S(M) = I(M), and the theorem is proved. 0 Theorem 42. Let A = {M 1 , M2, ... , M n , ... }, with M i E M 2 , and suppose there exist disjoint predictable sets Ai such that 1A'd[Mi , M i] = d[Mi , Mil, i 2:: 1. Let At = 2::1 f; lA, (s )d[Mi, MiJs. Suppose that (aJ E{Aoo } < 00; and (b) for:P C F oo such that for any Xi E bFi , we have Xl = E{XiIFt} = I~ H~dM;, t 2:: 0, for some predictable process Hi. Then M = 2::1 M i exists and is in M 2, and for any Y E b Vi?' if yt = E{YIFt}, we have that yt = f~ HsdMs, for the martingale M = 2::1 Mi and for some HE L(M). Proof. Let Nn = 2:~=1 Mi. Then [Nn, Nnh = 2:~=1 f; lA, (s)d[Mi , Mi]s, hence E{(N:;")2} = E{[Nn, Nn]oo} :::; E{Aoo }, and N n is Cauchy in M 2 with limit equal to M. By hypothesis we have that if Xi E bFi then E{XiIFt} = Xl = l t H;dM; = l t 1A,H;dM; = it 1A,H;d(M; + :L Ml) o #j = l t 1A,H~dMs. Therefore if i i=- j we have that [Xi,xj] = l t 1A,H~lAjHld[M,M]s = it 1A'nAj H;H1 d[M, M]s =0, since Ai n Aj = 0, by hypothesis. However using integration by parts we have 186 IV General Stochastic Integration and Local Times XIX; =Itx;_dx1 + it XLdX; + [Xi,xj]t = it Xi dXj + it Xl dXi s- s s- s o 0 = I t X;_1J1jH1dMs +I t XL1J1iH~dMs = it Hi,jdMs s, o where Hi,j is defined in the obvious way. By iteration we have predictable representation for all finite products rt 3 Martingale Representation 187 n M= ~Hi.Xi. i=l Next let (te)e>o be fixed times increasing to 00 with to = O. For each te we know there e~ist H i ,£ such that for a given locally square integrable local martingale N n Ntl'ltt = ~(Hi,e. Xi)tl'ltt' i=l By defining Hi = L~l H i,el(tt_l,tt] one easily checks that Hi E L(Xi) and that n Nt = LHi ·X i . i=l o Corollary 1. As in Theorem 43, let JF be the completed natural filtration of an n-dimensional Brownian motion. Then every local martingale M for JF is continuous. Proof In the proof of Theorem 43 we saw that the underlying probability law P is extremal for A = {Xl, ... ,Xn }. Therefore by Theorem 40(c), ev- ery uniformly integrable martingale is continuous. The corollary follows by stopping. 0 Corollary 2. Let X = (Xl, ... , xn) be an n-dimensional Brownian motion and let JF be its completed natural filtration. Then every local martingale M for JF has a representation where Hi are predictable. Proof By Corollary 1 any local martingale M is continuous, hence it is locally square integrable. It remains only to apply Theorem 43. 0 Corollary 3. Let X = (Xl, ... ,xn) be an n-dimensional Brownian motion and let JF be its completed natural filtration. Let Z E F(XJ be in Ll. Then there exist Hi predictable in L(Xi) with JO(XJ(H~)2ds< 00 a.s. such that 188 IV General Stochastic Integration and Local Times Proof. Let Zt = E{ZIFt}, taking the cadlag (and hence continuous) version. By Corollary 2 we have By Theorem 42 of Chap. II we have that Zt = B[z,z]t for some Brownian motion B, 0 < t < 00. Letting t tend to 00 shows that Zoo = Blimt~oo[Z,Zlt' which shows that [Z, Z]oo < 00 a.s. However, a.s. Finally, take t = 00, observe that Zoo = Z and that Fo is a.s. trivial. Hence Zo is constant and therefore Zo = E{Z}. 0 Corollary 4. Let X = (Xl, ... ,Xn ) be an n-dimensional Brownian motion and let JF be its completed natural filtration. Let Z E LI(Foo ) and Z > 0 a.s. Then there exist Ji predictable with Jooo (J;)2ds < 00 a.s. such that { n roo n 1 roo } Z = E{Z} exp 8 Jo J;dX; - ~"2 Jo (J;)2ds . Proof By Corollary 3 there exist predictable Hi such that if Zt = E{ZIFt}, then Therefore (Note that since (Zdt2:o is continuous and never 0, i is locally bounded.) The proof is completed by setting J; = is H; and taking exponentials of both sides. 0 Corollary 5. Let JF be the completed natural filtration of an n-dimensional Brownian motion. If T is a totally inaccessible stopping time, then T = 00 a.s. Proof This is merely Theorem 40 (a). o We end this section with a quite general martingale representation result, which while often true, nevertheless requires a countable number of martin- gales to have the representation. We then apply it to give a condition for compensators to have absolutely continuous paths. But first we state a defi- nition from measure theory that is not widely known. 3 Martingale Representation 189 Definition. A measurable space (0, F) is said to be separable if there exists a countable family of functions on (or subsets of) ° which generate the a- algebra :F. Theorem 44. Let (0, F, IF, P) be a jiltered complete probability space satisfy- ing the usual hypotheses. Assume also that the a-algebra F = F= and that it is separable. Then there exists a countable sequence of martingales (M1 , M 2 ,.··) in L 2 such that they are orthogonal, I:i>l E{[Mi, M i]=} < 00, and such that if N is any L2 martingale, then there -exists a sequence of predictable pro- cesses (HI, H 2 , ... ) such that Nt = I:i~l J; H;dM;. That is, the martingales {Mi, i 2': I} have martingale representation. Proof. Since F is separable, there exists a countable basis of L 2 , call it (MO,M 1,M2 , ...), where MO is constant and E{M i } = 0 for i 1= o. One can take such a basis to be orthonormal in the usual way. That is, E{MiMj} = 0 for i 1= j, and E{(M i )2} = 1 for all i. If one then multiplies Mi by 2-i and defines the martingales Mf = E{MiIFt}, we have the construction. The space of stochastic integrals with respect to all of the M i where the sum converges in L2 is easily seen to be a stable subspace. Next let N be any martingale in L2 with No = O. Then we have where L is orthogonal to the stable subspace generated by the countable collection of martingales (Mi)i~l. This implies that L= is orthogonal to all of the random variables M/x, = Mi. Since these random variables form a basis of L 2 , this implies that L must be constant. Since Lo = 0 we have that L is identically zero. 0 Before we apply Theorem 44 to compensators, we need to recall and for- malize two definitions, and prove Lebesgue's change of time formula. The first definition we have already seen in Exercise 7 of Chap. III, and the second one we saw in Theorem 42 of Chap. II as well as Exercises 31 through 36 of that chapter. Definition. A (right continuous) filtration IF on a filtered probability space satisfying the usual hypotheses is called quasi left continuous if for every predictable stopping time T one has FT = FT-. Note that if T is a predictable stopping time, and if Tn is an announcing sequence of stopping times for T, and if X E L 1 , then limn _= E{XIFTn } = E{XIFT -}. If M is a uniformly integrable martingale, then Therefore E{.6.MTIFT-} = 0 a.s. and if .6.MT is FT- measurable, it must be 0 almost surely. So if IF is quasi left continuous, no martingales can jump 190 IV General Stochastic Integration and Local Times at predictable times. Since the accessible part of any stopping time Can be covered by a countable sequence of predictable times, we conclude that if JF is quasi left continuous then martingales jump only at totally inaccessible stopping times. Definition. Let A = (Atk:~o be an adapted, right continuous increasing process, which need not always be finite-valued. The change of time (also known as a time change) associated to A is the process Tt = inf{s > 0: As > t}. Some observations are in order. We have that t f---> Tt is non-decreasing and hence Tt- exists. Also, since {At> s} = U,,>o{At > s + e}, we have that t f---> Tt is right continuous. It is continuous if A has strictly increasing paths. Moreover A Tt ~ A Tt _ ~ t, and Ah_)- ::::: A(Tt)- ::::: t. (Here we use the convention TO- = 0.) Finally note that {Ts - ::::: t} = {s ::::: Ad, which implies that Tt- is a stopping time, and since Tt = lim,,-;o T(t+,,)- we conclude that Tt is also a stopping time. Theorem 45 (Lebesgue's Change of Time Formula). Let a be a positive, finite, right continuous, increasing function on [0,00). Let c denote its right continuous inverse (change of time). Let f be a positive Borel function on [0,00). If G is any positive, finite, right continuous function on [0,00) with G(O-) = 0, then 100 f(s)dG(a(s» = 100 f(c(s- »l{c(s_) 3 Martingale Representation 191 Corollary. Let a be a positive, finite, continuous, strictly increasing function on [0, 00). Let c denote its continuous inverse (change of time). Let f be a positive Borel function on [0,00). Then l C(t) it° f(s)da(s) = ° f(c(s))ds. Proof It suffices to rewrite the left side of the equation as 100 l[o,c(t)] (s)f(s)da(s) and observe that c is also continuous and strictly increasing, which implies l[o,c(t)](c(s)) = l[o,t](s). 0 Our goal, stated informally in words, is if a filtration is quasi left contin- uous, then modulo a change of time, all compensators of adapted counting processes with totally inaccessible jumps have paths which are absolutely con- tinuous. This is achieved in Theorem 47. We begin with two definitions. Definition. Let (0, F,]F, P) be a filtered probability space satisfying the usual hypotheses, and let S denote the space of all square integrable martin- gales with continuous paths a.s. It is easy to see that S is a stable subspace. 12 For an arbitrary square integrable martingale M, let MC denote the orthogo- nal projection of M onto S. Then MC is called the continuous martingale part of M. If we write M = M C+ (M - M C ) as its orthogonal decomposition, then we have M d = (M -MC ) where M d is called the purely discontinuous part of the martingale M. Note that if a local martingale is locally square integrable, we Can extend the definition of continuous part and purely discontinuous part trivially, by stopping. Also note that the term "purely discontinuous" is misleading: it is not a description of a path property of a martingale, but rather simply refers to the property of being orthogonal to the stable subspace of continuous martingales. See Exercises 6 and 7 in this regard. Definition. Let (0, F,]F, P) be a filtered probability space satisfying the usual hypotheses. We call it an absolutely continuous space if for any purely discontinuous locally square integrable martingale M, d(M, Mh ~ dt. Theorem 46. Let (0, F,]F, P) be an absolutely continuous space. Then the compensators for all adapted counting processes with totally inaccessible jump times and without explosions, are absolutely continuous. Proof. Let N be aE adapted counting process and let N be its compensa~or, so that X = N - N is a locally square integrable local martingale. Since N is continuous we have [X, X]t = Ls~t(.6.Ns)2 = Nt, since (.6.Ns)2 = .6.Ns = 1 12 See Exercise 6. 192 IV General Stochastic Integration and Local Times when N jumps at s. Therefore N = {X, X), and since {X, X) has absolutely continuous paths by hypothesis, we are done. 0 Remark (Yan Zeng). With a little more work, one can show the following: Let (0, F, JF, P) satisfy the usual hypotheses. If it is an absolutely continuous space, then all jump times of square integrable martingales are totally inac- cessible. Moreover it is an absolutely continuous space if and only if for every totally inaccessible stopping time T, if Nt = l{t~T}, then dNt ~ dt a.s. If we take the uninteresting counting process Nt = I:i>1 l{t~i} which jumps a~ constant (and hence predictable) ti~es, then its compensator is simply Nt = Nt, and the martingale X = N -!Vis identically zero. To avoid these j;rivialities we assume that the process N is continuous. Note however that N is not a priori absolutely continuous. This is to explain our hypotheses in the next theorem: Theorem 47 (Absolutely Continuous Compensators). Let (O,F,JF,P) be a filtered probability space satisfying the usual hypotheses. Assume that the a-algebra F = F (XJ and that it is separable. 13 Assume further that JF is quasi left continuous. To prevent trivialities assume also that it is not an absolutely continuous space. Then there exists a change of time Tt and a new filtration G given by gt = F T " such that if N is a G adapted counting process with totally inaccessible jump times and without explosions, then dNt ~ dt. Proof. Let (Mi)i~1 be the countable sequence constructed in Theorem 44. Define the strictly increasing process At = t + I: i >dM i, Mi]t, and note that E{Ad < 00 for 0 :::; t < 00. Let C denote the compensator of A. One can check that Ct = t+ I:i>1{Mi,Mik Then C is both strictly increasing and continuous, since JF is quasi left continuous. Let Tt = inf{s > 0 : Cs > t} and then Ct = inf{s > 0: Ts > t}, where T is the time change of the theorem statement. Let N be a G adapted counting process and let X = N - N. As- sume, by stopping if necessary, that X(XJ E L 2 . We write {X, X) G to de- note the sharp bracket taken using the G filtration. Note that L t = Nc, is also a counting process, and since T continuous implies that gc, C F t (see Exercise 38 of Chap. II), we conclude L is adapted to JF. We then have [L, L]t = I: s 4 Martingale Duality and Jacod-Yor Theorem 193 and by the uniqueness of the compensator of L we have {X, X)g, = {L, L)f. On the other hand, by Theorem 44 we know that Lt - It = I:i~l J; H;dM; for some predictable processes (Hi), and thus {L - I, L - I)r = L rt(H;)2d{Mi,Mi)~ = L rt(H;)2c~dCsi~l io i~l io = l t JsdCs for some (predictable) nonnegative processes ci whose existence is assured because d{Mi, Mi)t « dCt a.s. The process Js = I:i~l (H;)2C~, by Fubini's Theorem. Next, note that since {L-I, L-I)r = {X, X)g" we have {X, X)f = (L - I, L - I)~, = J;' JsdCs, which by the corollary of Lebesgue's change of time formula, implies {X, X)f = J; JTsds. 0 4 Martingale Duality and the J acod-Yor Theorem on Martingale Representation We have already defined the space li2 for semimartingales; we use the same definition for martingales, which in fact historically Came before the definition for semimartingales. We make the definition for all p, p ~ 1. Definition. Let M be a local martingale with cadlag paths. We define the liP norm of M to be IIMllp = E{[M, M]~2p/p. If IIMlhtP < 00, we say that M is in liP. Finally we call liP the space of all ca. 194 IV General Stochastic Integration and Local Times We have the following simple results concerning 'Hi. Theorem 49. 'H2 C 'Hi and local martingales of integrable variation are a subset of 'Hi. Proof. First note that if M is a cadlag martingale, then E{ J[M,M]oo} :::; (E{[M, M]oo})1/2 which gives the first statement. For the second, if M has integrable variation then [M,M]oo = Ls(.6.Ms)2 :::; CLs I.6.Ms I)2, whence J[M, M]oo :::; Jooo IdMsl, and taking expectations, we have the result. 0 Theorem 50. 'H2 is dense in 'Hi and bounded martingales are dense in 'Hi. Proof. Let M be a local martingale in 'Hi. By the Fundamental Theorem of Local Martingales (Theorem 25 of Chap. III) we have M = N + U where N is a local martingale with jumps bounded by 1 and U has paths of locally integrable variation. If (Tn )n>l is a sequence of stopping times increasing to 00 a.s., then I/MTnl/'}-£l :::; IIMI/;l, and also MTn converges to M in 'Hi. Choose Tn to be the first time [U, U]t has total variation larger than n, and also I[N,NjTnl:::; n-l; that is, Tn = inf{t ~ 0: J~[U,U]t > n, or [N,NJt > n-l} with Tn increasing to 00. Then [N, N]Tn :::; n and thus NTn is in 'Hi for each n. Therefore we turn out attention to UTn, and since [U, UJTn :::; n + I.6.UTn I, we need consider only the one jump of U at time Tn: .6.UTn. Let ~ denote .6.UTn ' Letting ~k = ~l{I~I~k}' since ~k is bounded we Can compensate ~kl{t~Tn}' and call the resulting martingale Vt. The compensator of ~kl{t~Tn} is in 'H 2 because ~k is bounded. We have then that V; is in 'H 2 and converges to Vn as k ---> 00 in total variation norm, and hence also in 'Hi. This proves that 'H2 is dense in 'Hi, and since bounded martingales are dense in 'H 2 , they too are dense in 'Hi. 0 Theorem 51. Let M be a local martingale. Then M is locally in 'Hi. Proof. By the Fundamental Theorem of Local Martingales we know that M = N + U, where N has jumps bounded by a constant /3 and U is locally of integrable variation. By stopping, we thus assume that N is bounded and U has paths of integrable variation. The result then follows by the previous theorem (Theorem 49). 0 One can further show, by identifying 'HP with LP through the terminal value of the martingale, that 'HP is a Banach space (in particular it is complete) for each p ~ 1. One can further show that the dual space of continuous linear functional on 'HP is 'Hq , where ~ + *= 1, 1 < p < 00. This continues nicely the analogy with LP except for the case p = 1. It can be shown (see [47]) that the dual of 'Hi is not 'Hoo . It turns out that a better analogy than LP is that of Hardy spaces in complex analysis, where C. Fefferman showed that the dual of 'Hi can be identified with the space of functions of bounded mean oscillation, known by its acronym as BMO. With this in mind we define the space of BMO martingales. 4 Martingale Duality and Jacod-Yor Theorem 195 Definition. Let M be a local martingale. M is said to be in BMO if M is in fi2 and if there exists a constant c such that for any stopping time T we have E{(Moo - Mr_)2IFr} ::::: c2 a.s., where Mo- = 0 by convention. The smallest such c is defined to be the BMO norm of M, and it is written IIMIIBMo. If the constant c does not exist, or if M is not in fi2, then we set IIMIIBMO = 00. Note that in the above definition E{M~IFo} ::::: c2 (with the convention that M o- = 0) and therefore t1MII1-l2 ::::: IIMIIBMo. Note in particular that IIMltBMO = 0 implies that M = O. Let T be a stopping time and A E Fr. Replacing T with TA shows that the above definition is equivalent to the statement for every stopping time T. This in turn gives us an equivalent description of the BMO norm: IIMIIBMO = sup T E{ (Moo - Mr - )2} P(T < (0) where the supremum is taken over all stopping times T. Note that this second characterization gives that II . tlBMo is a semi-norm, since it is the supremum of quadratic semi-norms. An elementary property of BMO martingales is that ME BMO if and only if all of the jumps of M are uniformly bounded. Thus trivially, continuous L 2 martingales and bounded martingales are in BMO. The next inequality is quite powerful. Theorem 52 (Fefferman's Inequality). Let M and N be two local mar- tingales. Then there exists a constant c such that Fefferman's inequality is a special case of the following more general result. Theorem 53 (Strengthened Fefferman Inequality). There exists a con- stant c such that for all local martingales M and N, and U an optional process, Proof. Let Ct = J~ U;d[M, M]s and define Hand K by 2 U? 2 y75; H t = va; + JCt_1{t>o} l{ct >o}, K t = Ct. 196 IV General Stochastic Integration and Local Times Using integration by parts yields 2 d~ ~Ht d[M, M]t = I{Ct >O} va; JC = l{t>O}dV Ct· t + t- 1{t>O} From the definitions of Hand K we have 2 2 1 2Ht Kt ~ "2Yt I{C,>o}· The Kunita-Watanabe inequality implies 100 IUsll{cs =o}ld[M,N]sl ~ (1 00 U;1{Cs=o}d[M,M]s)1/2([N,N]00)1/2 = (1 00 I{Cs=o}dCs)1/2([N, N]00)1/2 = 0 a.s., and since Id[M, N]s I is absolutely continuous with respect to d[M, M]s as a consequence of the Kunita-Watanabe inequality, we have 1 roo 1 roo v'2 E{Jo IUslld[M,N]sl} = v'2 E{Jo 11{cs>o}Uslld[M,N]sl} ~ E{l°O IHsKslld[M,N]sl} r-------- < E{l°O H;d[M,M]s} E{l°O K;d[N,N]s}. But E{l°O H;d[M, M]s} ~ E{l°O dVC:-} = E{ Jcoo } = E{(l°O U;d[M, M]s)1/2} and E{100 K;d[N, N]s} = E{100 ([N, N]oo - [N, N]s- )dK;} = E{l°O (E{[N,N]ooIFs} - [N,N]s_)dK;}. But E{E{[N,N]ooIFs} - [N,N]s-} is bounded by IINII~Mo on (0,00), hence we have that E{l°O K;d[N,N]s} = E{l°O (E{[N,N]ooIFs} - [N,N]s_)dK;} ~ IINII~MOE{JCoo }, and the result follows. o 4 Martingale Duality and Jacod-Yor Theorem 197 Remark. The constant c in Theorems 52 and 53 can be taken to be V2, as can be seen from an analysis of the preceding proof. Theorem 54. Let N E }{2. Then N is in BMO if and only if there is a constant c > 0 such that for all M E }{2, IE{[M,N]oo}1 ~ cllMllw· Moreover IINIIBMO :::; V6c. Proof If N is in BMO, then we can take c = V2IINIIBMO from Fefferman's inequality (Theorem 52) and the remark following the proof of Theorem 53. Now suppose that M is in}{2 and that IE{[M,N]oo}1 :::; cllMllw; we want to show that N is in BMO. We do this by first showing that INol ~ c a.s., and then showing that N has bounded jumps. Let A = {INol > c}. Suppose P(A) > O. Let ~ = Si~/~o) lA. Then E{I~I} = 1, and if we define the trivial martingale M t = ~ for all t ::::: 0, then M E }{2 and IIMllw = E{I~I} = 1, whence IE{[M,N]oo}1 = E{MoNo} = E{ IN~il{llol>c;}> c = cllMllw.P No >c This of course is a contradiction and we conclude INol :::; c a.s. We next show IL).NI :::; 2c. Since every stopping time T can be decomposed into its accessible and totally inaccessible parts, and since each accessible time can be covered by a countable collection of predictable times with disjoint graphs, we can assume without loss of generality that T is either totally inac- cessible or predictable. We further assume P(T > 0) = 1. Suppose then that P(IL).NTI > 2c) > 0, and set sign(L).NT)~ = P(IL).NTI > 2c) 1{I~NTI>2c}· Let M be the martingale consisting of ~1{t~T} minus its compensator. Then M is in }{2 and has at most one jump, which occurs at T. The jump is given by L).M = {~ - E{~IFT-}, T is predictable, T ~ T is totally inaccessible. Note that we also have IIMllw is less than the expected total variation of M, which in turn is less than 2E{I~I} = 2. If T is totally inaccessible then E{[M N] } = E{L).M L).N } = E{cL).N } = E{ IL).NTll{I~NTI>2C}} , 00 T T 2c) > 2c ::::: c IIMllw, which is a contradiction. On the other hand, if T is predictable, then we know that E{L).NTIFT-} = °and thus we are reduced to the same calculation and 198 IV General Stochastic Integration and Local Times the same contradiction. We conclude that P(I.6.NT I > 2c) = 0, and thusN has jumps bounded by 2c. Last let T be any stopping time. Let M = N - NT and "I = [N, N]oo - [N, N]T. Then M is in H2 and [M, M]oo = [M, N]oo = "I. By our hypotheses it now follows that E{TJ} = E{[M,N]oo} ~ c IIMII1i1 = cE{~} = cE{J1]l{T 0 : INti ~ n}. Then INTn I :s; n + c, and N is locally bounded. 0 The key result concerning HI and BMO is the Duality Theorem which is Theorem 55 that follows. First let us lay the foundation. For N chosen and fixed in BMO we define the operator LN from HI to lR by for all M in HI. Then one can easily check to see that LN is linear, and Feffer- man's inequality proves that it is bounded as well, and therefore continuous. If BMO is the Banach space dual of HI then it is also complete, a fact that is apparently not easy to verify directly. 4 Martingale Duality and Jacod-Yor Theorem 199 Theorem 55 (The Dual of r£l is BMO). The Banach space dual of all (bounded) linear functionals on HI can be identified with BMO. Moreover if LN is such a functional then the norms IILNII and IINIIBMO are equivalent. Proof. Let N be in BMO. By Fefferman's inequality we have for all M in HI. This shows that LN is in the dual of HI and also that IILNII :s; c IINIIBMO. Note further that LN cannot be trivial since LN(N) = E{[N, N]oo} > 0 unless N is identically O. Therefore the mapping F(N) = L N is an injective linear mapping from BM0 into H h , the dual of HI. Let L be an arbitrary linear functional in the dual of HI. We have IL(M)I :s; IILIIIIMII1t1 :s; IILIIIIMllw· This means that L is also a bounded linear functional on H 2 . Since H 2 is isomorphic as a Hilbert space to the L 2 space of the terminal random variables of the martingales in H 2 , we have that there must exist a unique martingale N in H 2 such that for any M in HI we have: Clearly LN = L on H 2 , and since H 2 is dense in HI by Theorem 50, we have that Land LN are the same functional on HI. This shows that BMO equipped with the norm IILNII is isomorphic to H h and thus, being the dual of a Banach space, it is itself a Banach space and in particular it is complete. Combining equation (*) with Theorem 54 we have that IILNII and IINIIBMO are equivalent norms. This completes the proof. 0 While Fefferman's inequality, the space of BMO martingales, and the du- ality of H oo and BMO are all of interest in their own right, we were motivated to present the material in order to prove the important Jacod-Yor Theorem on martingale representation, which we now present, after we recall the version of the Hahn-Banach Theorem we will use. Theorem 56 (Hahn-Banach Theorem). Let X be a Banach space and let Y be a closed linear subspace. Then Y = X if and only if the only bounded linear functional L which has the property that L(Y) = ° is the functional which is identically zero. Theorem 57 (Jacod-Yor Theorem on Martingale Representation). Let A be a subset of H 2 containing constant martingales. Then S(A), the stable subspace of stochastic integrals generated by A, equals H 2 if and only if the probability measure P is an extremal point of M 2 (A), the space of probability measures making all elements of A square integrable martingales. 200 IV General Stochastic Integration and Local Times Proof The necessity has already been proved in Theorem 38. By the Hahn- Banach Theorem, HI = S(A) if and only if L(S(A)) = 0 implies L is identi- cally zero, where L is a bounded linear functional. Let L be a bounded linear functional which is such that L(S(A)) = O. Then there exists a martingale N in BMO such that L = LN. The local martingale N is locally bounded, so by stopping we can assume it is bounded and that No = O. (See the two corollaries of Theorem 54.) Let us also assume it is not identically zero, and let c be a bound for N. We can then define two new probability measures Q and R by dQ = (1- ~;)dP, dR = (1 + ~;)dP. Then Q and R are both in M 2 (A), and P = !Q + !R shows that P is not extremal in M 2 (A), a contradiction. Therefore we must have that L is identically zero and we have that HI = S(A). As far as H 2 is concerned, it is a subspace of HI, hence H 2 C S(A). But by construction of S(A), it is contained in 1{2, and we have martingale representation. 0 5 Examples of Martingale Representation In Sect. 3 we have already seen the most important example of martingale representation, that of Brownian motion. In this section we give a method to generate a family of examples which are local martingales with jumps, and which have the martingale representation property. The limitations are that the family of examples is one dimensional (so that we exclude vector- valued local martingales such as n-dimensional Brownian motion), and that the descriptions of the jumps are all of the same rather simple kind. The idea is to construct a class of local martingales H such that for X E 1{ we have both that X o = 0 and the compensator of [X, X]t is At = t. If X has the martingale representation property, then there must exist a predictable process H such that The above equation is called Emery's structure equation, and it is written (in a formal sense) in differential notation as In order to establish that solutions to Emery's structure equation actually exists we write it in a form resembling a differential equation: d[X, X]t = dt + ¢(Xt- )dXt Equation (**) is unusual and different from the stochastic differential equa- tions considered later in Chap. V, since while the unknown is of course the 5 Examples of Martingale Representation 201 local martingale X (and part of the structure equation is to require that any solution X be a local martingale), no a priori stochastic process is given in the equation. That is, it is lacking the presence of a given stochastic driving term such as, for example, a Brownian motion, a compensated Poisson pro- cess, or more generally a Levy process. Since therefore no probability space is specified, the only reasonable interpretation of equation (**) is that of a weak solution. That is, we want to show there exists a filtered probabil- ity space (0, F, JF, P) satisfying the usual hypotheses, and a local martingale X, such that X verifies equation (**). It would also be nice to have weak uniqueness which means that if X and Yare solutions of (**) for a given ¢, possibly defined on different filtered probability spaces, then X and Y have the same distribution as processes. That means that for every A, a Borel set on the function space of dl,dlag functions mapping JR.+ to JR., we have P(w : t f-7 Xt(w) E A) = Q(w : t f-7 Xt(w) E A), where P and Q are the probability measures where X and Yare respectively defined. Inspired by knowledge of stochastic differential equations, it is natural to conjecture that such weak solutions exist and are unique if the coefficient ¢ is Lipschitz continuous. 14 This is true for existence and was proven by P. A. Meyer [179]; see alternatively [136]. Since the proof uses weak conver- gence techniques which are not within the scope of this book, we omit it. Theorem 58 (Existence of Solutions of the Structure Equation). Let ¢ : JR. ---+ IR be Lipschitz continuous. Then Emery's structure equation has a weak solution with both (Xtk~o and (J~ ¢(Xs-)dXsk::o local martin- gales. The issue of uniqueness is intriguing. Emery has shown that one has uniqueness when ¢ is linear, but uniqueness for others ¢'s, including the Lip- schitz case, is open. The next theorem collects some elementary properties of a solution X. Theorem 59. Let X be a (weak) solution of (* **). Then the following hold. (i) E{Xn = E{[X,X]t} = t, and X is a square integrable martingale on compact time sets. (ii) All jumps of X are of the form l:i.Xt = ¢(Xt-). (iii) X has continuous paths if and only if ¢ is identically 0, in which case X is standard Brownian motion. (iv) If a stopping time T is a jump time of X, then it is totally inaccessible. Proof We prove the statements in the order given. Since a solution X and the integral term are both required to be local martingales, we know 14 Lipschitz continuity is defined and discussed in Chap. V. 202 IV General Stochastic Integration and Local Times there exists a sequence (Tn )n>l of stopping times increasing to 00 such that J;I\Tn ¢(Xs-)dXs is in £1. T~refore E{[X,XltI\Tn} = E{tI\Tn }, and apply- ing the Monotone Convergence Theorem to each side of the equality in this equation yields E{[X, X]d = t which further implies that X is a martingale on [0, tj for each t < 00 and that E{xi} = E{ [X, X]d = t. For the second statement, recall that [X,Xlt = [X,X]f + ~)~Xs)2, s5,t and hence we have ~[X,X]t = (~Xt)2 = ¢(Xt-)~Xt, and dividing both sides by ~Xt (when it is not zero) gives the result. For the third statement, suppose ¢ is identically zero. Then by the second statement X has no jumps and must be continuous. Since X is then a con- tinuous local martingale with [X, X]t = t, it is Brownian motion by Levy's Theorem. For the converse, if we know that X is continuous, if it is non-trivial it has paths of infinite variation since it is a local martingale. Thus so too does the term J; ¢(Xs-)dXs. But notice that this term is the right side of equa- tion (* * *) which is of finite variation, and we have a contradiction, so we must have that ¢ is zero. For the fourth statement, we implicitly stop X so that it is a uniformly integrable martingale. Next let T be a jump time of X. Then T = TA 1\ TB where A is the accessible part of T and B is the totally inaccessible part of T. Since TA can be covered by a countable sequence of predictable times with disjoint graphs, we can assume T A is predictable without loss of generality. Thus it suffices to show P(A) = 0. However since TA is predictable, and X is a uniformly integrable martingale, we have E{~XTA IFTA _} = 0. But ~XTA = ¢(XTA-) by part (ii) of this theorem, and ¢(XTA _) EFTA _ since T A is predictable, which implies that the jump of X at T A is zero, which in turn implies that P(A) = O. 0 Let us now consider a special class of structure equations where ¢ is assumed to be an affine function. That is, we assume ¢ is of the form ¢(x) = a + f3x. We analyze these special cases when a and f3 vary. Emery has named the solutions corresponding to affine structure equations the Azema martingales, since J. Azema's work on Markov processes and expansion of filtrations led him to the amazing formula of "the" Azema martingale given later in Sect. 7. Equation (**) now becomes and when f3 = 0, it reduces to d[X, Xlt = dt + adXt , and when in addition a = 0 we have seen that X is standard Brownian mo- tion. Note that Levy's TheOrem gives us weak uniqueness in this case, since 5 Examples of Martingale Representation 203 any solution with 0: = /3 = 0 must have the same distribution, namely that of Wiener measure. We have much more, as we see in the next theorem. However it has a long and difficult proof. Rather than present it, we refer the inter- ested reader to the excellent treatment of M. Emery, in his original paper [69] proving the result. Theorem 60 (Emery's Uniqueness Theorem). Let X be a local martin- gale solution of the structure equation d[X,X]t = dt + (0: + /3Xt-)dXt , X o = xo. Then X is unique in law. That is, any other solution Y must have the same distribution as does X. Moreover X is a strong Markov process. The uniqueness is especially significant in light of the next theorem. By martingale representation we mean that every square integrable martin- gale can be represented as a stochastic integral with respect to one fundamen- tal local martingale. Theorem 61. Consider the equation (0) on a filtered probability space (0, F, JF, P) which satisfies the usual hypotheses. Then X has martingale representation for its completed natural filtration if and only if the law P is an extreme point of the convex set of all probabilities on (0, F, JF) for which X is a martingale and verifies the equation. Moreover if the equation with fixed initial condition X o has weak uniqueness of solutions, then every solution X of the equation has martingale representation with respect to the smallest filtration satisfying the usual hypotheses and to which X is adapted. Proof. By the Jacod-Yor Theorem (Theorem 57) we need to verify that Pis extremal in the set M 2 of all probability measures such that X is a square integrable martingale. It is clearly true if P is extremal. Suppose then that P is not extremal, and let Q and R both be in M 2 , such that P = AQ+ (1- A)R, with 0 < A < 1. Both Q and R are absolutely continuous with respect to P, so under Q and R the terms [X,X] and J~ ¢(Xs-)dXs are the same a.s. (resp. dQ and dR). Therefore X satisfies equation (0) for both Q and R as well as P. Thus if P is not extremal in M 2 then it is also not extremal in the set of probability measures such that X satisfies equation (0). To prove the second statement, let 0 denote the canonical path space of cadlag paths, with X being the projection process given by Xt(w) = w(t). We have just seen that among the solutions of equation (0), the ones having mar- tingale representation are those whose law constitutes an extremal probability measure. But if weak uniqueness holds, the collection of all such probabilities consists of only one, and thus extremality is trivial. 0 204 IV General Stochastic Integration and Local Times We now examine several special cases as a and f3 vary. Let a = f3 = 0, and we have seen that X is standard Brownian motion. Because of Theo- rem 61 we conclude that (one dimensional) Brownian motion has martingale representation, recovering a special case of Theorem 43. Next suppose a = 1 and f3 = O. In this case the equation of Theorem 60 with X o = 0 becomes, in integral form, [X, X]t = t + (Xt - Xo) = t + X t . Therefore X t = [X, X]t - t and hence X is a finite variation martingale. Moreover l:1Xt = 1, so X only jumps up, with jumps always of size 1. Now let N be a standard Poisson process with arrival intensity A = 1, and let X t = Nt-t, the compensated Poisson process. Then X satisfies the equation, and by weak uniqueness all such X are compensated Poisson processes with A = 1. We conclude that a compensated standard Poisson process has martingale representation with respect to its natural (completed) filtration. For general a (and not just a = 1), but still with f3 = 0, it is simple to check that X;: = a(N t - ..!-2) ;:;'r a is the unique solution of the equation of Theorem 60 if N is a standard Poisson process. Note that (as is well known) XO: converges (weakly) to Brownian motion as a --+ o. We now consider the more interesting cases where f3 I- o. We repeat the equation of Theorem 60 here (in integrated form) for ease of reference: Observe that X is a solution of the above equation if and only if X + ~ is a solution of d[X,X]t = dt + f3Xt_dXt (00) with initial condition Xo = Xo + ~. Therefore without loss of generality we can assume that a = 0 and we do this from now on. We have two explicit examples for equation (00). When f3 = -1, we take where M is Azema's martingale15 , B is standard Brownian motion, and gt = sup{s :::; t : Bs = O}. By Theorem 86 of Sect. 8 of this chapter, we know that [X,X]f = 0 and [X,X]t = 2gt . Integration by parts gives J; Xs_dXs = (t - gt) - gt = t - [X,X]t, since [X,X]t = L s 6 Stochastic Integration Depending on a Parameter 205 Mis Azema's (original) martingale as presented in Sect. 8, and thus we have martingale representation for Azema 's martingale. Our second (and last) explicit example is for f3 = -2. In this case our equation becomes 1 P(Xt = vt) = P(Xt = -vt) = -2 for all t > O. The jumps of X occur from a change of sign, and they arrive according to a Poisson process with intensity "itdt. Such a process can be seen to be a martingale (as in [204]) by constructing a process X with a distribution as above and with filtration JF. Then for 0 < s < t, [X,X]t - t = -21t Xs_dXs and using integration by parts we obtain 2 J~ Xs_dXs = Xl- [X, XJt. Equat- ing terms gives Xl = t, and since E{Xt } = 0 for all t because XQ = 0 we deduce that where N is independent of the (i-algebra F s and has a Poisson distribution with parameter A = ! In( ~ ). In this way it is easily seen to be a martingale. We call it the parabolic martingale. Once again we are able to conclude: we have martingale representation for the parabolic martingale. 6 Stochastic Integration Depending on a Parameter The results of this section are of a technical nature, but they are needed for our subsequent investigation of semimartingale local times. Nevertheless they have intrinsic interest. For example, Theorems 64 and 65 are types of Fubini Theo- rems for stochastic integration. A more comprehensive treatment of stochastic integration depending on a parameter can be found in Stricker-Yor [219] and Jacod [103]. Throughout this section (A, A) denotes a measurable space. Theorem 62. Let yn(a, t, w) be a sequence of processes that are (i) A 18> B(IR+) 18> F measurable, and (ii) for each fixed a the process yn(a, t,w) is cddldg. Suppose yn (a, t, .) converges in ucp for each a E A. Then there exists an A 0 B(IR+) 0 F measurable process Y = Y(a, t,w) such that (a) Y(a, t,') = limn-ooo yn(a, t,') with convergence in ucp; (b) for each a E A, Y is a.s. cddldg. Moreover there exists a subsequence nk (a) depending measurably on a such that limnk(a)-ooo ~nk(a) = yt uniformly in t on compacts, a.s. 206 IV General Stochastic Integration and Local Times Proof. Let S~,i,j = SUPt:c;u Iyi(a, t,·) - yj(a, t, .)[. Since yi is cadlag in t the function (a,w) I----' S~,i,j is A 0 F measurable. By hypothesis we have limi,j->oo S~,i,j = 0 in probability. Let no(a) = 1, and define inductively nk(a) = inf{m > max(k, nk-l(a)): sup P(Si: i j > T k) ~ T k}. i,j?:m ' l We then define Zk(a t w) = ynk(a)(a t w), , " . Since each a I----' nk (a) is measurable, so also is Zk. Define T:: i j = sup IZi(a, t,w) - Zj(a, t,w)l; , , t'S:u then also (a, w) I----' T::, i,j (w) is jointly measurable, since Zi have cadlag paths (in t). Moreover by our construction P(Tk,k,k+m > 2-k) ~ 2-k for any m 2: 1. The Borel-Cantelli Lemma then implies that limi,j->oo T::',i,j = 0 almost surely, which in turn implies that lim Zi(a, t,·) exists a.s.,2->00 with convergence uniform in t. Let Aa be the set where Zi converges uniformly (note that Aa E A 0 F and p(Aa) = 1, each fixed a), and define Y( ) _ {limi->oo Zi(a, t,w), wE Aa,a,t,w - 0, w!j. Aa. Then Y is cadlag thanks to the uniform convergence, and it is jointly mea- surable. 0 Theorem 63. LetX be a semimarlingale withXo = 0 a.s. and let H(a, t,w) = Hf(w) be A0P measurable16 and bounded. Then there is a function Z(a, t,w) in A 0 B(IR+) 0 F such that for each a E A, Z(a, t,w) is a cddldg, adapted version of the stochastic integral J; H~dXs. Proof. Let H = {H E bA0P such that the conclusion of the theorem holds}. If K = K(t,w) E bP and f = f(a) E bA, and if H(a,t,w) = f(a)K(t,w), then I t H(a, s, ·)dXs = I t f(a)K(s, ·)dXs = f(a) I t K(s, ·)dXs, and thus clearly H = f K is in H. Also note that H is trivially a vector space, and that H of the form H = f K generate bA 0 P. Next let H n E H and suppose that H n converges boundedly to a process H E bA 0 P. By Theorem 32 (for example) we have that Hn . X converges uniformly in t in probability on compacts, for each a. Therefore H E H, and an application of the Monotone Class Theorem yields the result. 0 16 Recall that P denotes the predictable iT-algebra. 6 Stochastic Integration Depending on a Parameter 207 Corollary. Let X be a semimartingale (Xo = 0 a.s.), and let H(a, t, w) = Hf(w) E A 0 P be such that for each a the process Ha E L(X). Then there exists a function Z (a, t, w) = Zf E A 0 B(lR+) 0:F such that for each a, Zf is an a.s. cadlag version of f~ H~dX8' Proof By Theorem 32 the bounded processes za,k = Hal{IHal::;krX converge to Ha .X in ucp, each a. But za,k can be chosen cadlag and jointly measurable by Theorem 63. The result now follows by Theorem 43. 0 Theorem 64 (Fubini's Theorem). Let X be a semimariingale, Hf = H (a, t, w) be a bounded A 0 P measurable function, and let J.l be a finite mea- sure on A. Let Zf = f~ H~dX8 be A 0 B(lR+) ®:F measurable such that for each a, za is a cadlag version of H a . X. Then yt = fA ZfJ.l(da) is a cadlag version of H· X, where Ht = fA HfJ.l(da). Proof. By pre-stopping we may assume without loss of generality that X E 1-£2, and because the result holds for the finite variation part of the canonical decomposition of X by the ordinary Stieltjes Fubini Theorem, we may further assume that X is a martingale with E{[X, X]oo} < 00. Next suppose Hf is of the form H(a,t,w) = K(t,w)f(a) where K E bP and f is bounded, measurable. Then K E L(X) and f If(a)IJ.l(da) < 00. In this case we have Zf = f(a)K' X, and moreover JZfJ.l(da) = Jf(a)K· XJ.l(da) = K· X Jf(a)J.l(da) = (J f(a)J.l(da)K) . X =H·X. By linearity the same result holds for the vector space V generated by pro- cesses of the form K(t, w)f(a) with K E bP and f bounded, measurable. By the Monotone Class Theorem it now suffices to show that if H n E V and limn .....oo Hn = H, then the result holds for H. Let Z~,t = H~ . X, the cadlag version. Then by Jensen's and the Cauchy-Schwarz inequalities, II~II (E{L s~p IZ~,t - ZflJ.l(da)})2 ~ E{ rsup IZ~ t - ZfI 2J.l(da)} = rE{sup IZ~ t - ZfI 2}J.l(da)JA t' JA t ' ~ 4 L E{(Z~,oo - Z~)2}J.l(da) = 4 L E{[Z~ - za,z~ - za]oo}J.l(da) by Doob's quadratic inequality for the martingales Z~ and za, and by Corol- lary 3 of Theorem 27 of Chap. II. Continuing, the preceding equals 208 IV General Stochastic Integration and Local Times and the above tends to 0 by three applications of the Dominated Convergence Theorem. We conclude from the preceding that rsup IZ~,t - Z~IJl(da) < ()() a.s.JA t and therefore fA IZfIJl(da) < ()() for all t, a.s. Moreover E{sup I r Z~ tJl(da) - r ZfJl(da)I}:s: E{ r sup IZ~,t - ZfIJl(da)} t JA' JA JA t which tends to O. Therefore taking Hn,t = f H~,tJl(da) we have Hn . Xt = fA Z~,tJl(da) converges in ucp to f ZfJl(da). Since Hn · X converges to H· X by Theorem 32, we conclude H· X = f ZfJl(da). 0 The version of Fubini's Theorem given in Theorem 64 suffices for the applications of it used in this book. Nevertheless it is interesting to determine under what more general conditions a Fubini-type theorem holds. Theorem 65 (Fubini's Theorem: Second Version). Let X be a semi- martingale, let Hf = H (a, t, w) be A 0 P measurable, let Jl be a finite positive measure on A, and assume Letting Zf = f~ H:dXs be A 0 B(lR+) ®:F measurable and za cadlag for each a, then yt = fA ZfJl(da) exists and is a cadlag version of H· X, where Ht = fA HfJl(da). Proof. By pre-stopping we may assume without loss of generality that X E H 2 and that IIHaIIL2(dl-') is (H 2 ,X) integrable. Let X = N + A be the canonical decomposition of X. Then Next observe that E{l°O IIH:IIL2(dl-')ldAs l} ~ E{"l°O IH~lldA811IL2(dl-')} ~ cE{"l°O IH~lldA811ILl(dl-')} = ci E{l°O IH~lldAsl}Jl(da). Also 6 Stochastic Integration Depending on a Parameter 209 E{l°OL(H~)2 jl(da)d[N, N]s} = LE{l°O (H~)2d[N,N]s}jl(da), and therefore E{jooo IH~lldAsl} < 00 and E{jooo(H~)2d[N,N]s} < 00 for jl-almost all a E A. Whence H a E L(X) for jl almost all a E A. Next define H n = H1{IHj::;n}, and the proof of Theorem 64 works as well here. 0 The hypotheses of Theorem 65 are slightly unnatural, since they are not invariant under the transformation H -7 _1_Ha ip(a) where ip is any positive function such that f ip(a)jl(da) < 00. This can be alleviated by replacing the assumption (fA (Ha)2jl(da)) 1/2 E L(X) with (W)2(f 0 for all a E A. Let X be standard Brownian motion, let to < t1 < t2 < ... be an increasing sequence in [0,1], and define Then is in L2(dt), whence Ht = fA IHfljl(da) E L(X), and moreover if t ~ 1, ~ 1 -1/2 )H· X t = 0 -(ta - ta-t) (Xta - X ta _ l ,a a=l where the sum converges in L 2 • However if t ~ 1 then 210 IV General Stochastic Integration and Local Times and because (ta -ta_d-1/2(Xta -Xta _,) is an Li.d. sequence and 2:::::1 a-1 = 00. Note that this example can be modified to show that we also cannot replace the assumption that (L (H:)2Jl(da)) 1/2 E L(X) with the weaker assumption that UA(Hf)PJl(da))l/ p E L(X) for some p < 2. 7 Local Times In Chap. II we established Ito's formula (Theorem 32 of Chap. II) which showed that if f : IR -> IR is C2 and X is a semimartingale, then f(X) is again a semimartingale. That is, semimartingales are preserved under C2 transfor- mations. This property extends slightly: semimartingales are preserved under convex transformations, as Theorem 66 below shows. (Indeed, this is the best one can do in general. If B = (Bt)t>o is standard Brownian motion aqd yt = f(Bt ) is a semimartingale, then f must be the difference of convex func- tions. (See Qinlar-Jacod-Protter-Sharpe [34].) We establish a related result in Theorem 71, later in this section.) Local times for semimartingales appear in the extension of Ito's formula from C2 functions to convex functions (Theo- rem 70). Theorem 66. Let f : lR -> IR be convex and let X be a semimartingale. Then f(X) is a semimartingale and one has where f' is the left derivative of f and A is an adapted, right continuous, increasing process. Moreover LlAt = f(Xt ) - f(Xt-) - f'(Xt-)LlXt . Proof First suppose IXI is bounded by n, and in re, and that Xo = O. Let 9 be a positive Coo function with compact support in (-00,0] such that J~oo g(s)ds = 1. Let fn(t) = n J~oo f(t + s)g(ns)ds. Then fn is convex and C2 and moreover f~ increases to f' as n tends to 00. By ItO's formula 7 Local Times 211 where A~ = L {fn(Xs) - !n(Xs-) - !~(Xs-)boXs} + ~ it j::(Xs_)d[X,X]~. Ooo A~ = At in L2, and where the convergence of the stochastic integral terms is in 1{2 on [0, t]. We now compare the jumps on both sides of the equation (*). Since f~ !'(Xs-)dXs = 0 we have that A o = O. When t > 0, the jump of the left side of (*) is j(Xd - j(Xt-), while the jump of the right side equals j'(Xt_)boXt + boAt. Therefore boAt = j(Xt ) - j(Xt-) - !'(Xt-)boXt , and the theorem is established for IXI bounded by n and in ,.e. Now let X be an arbitrary semimartingale with Xo = O. By Theorem 13 we know there exists a sequence of stopping times (Tn)n>l' increasing to 00 a.s. such that X Tn - E ,.e for each n. An examination of the proof of Theorem 13 shows that there is no loss of generality in further assuming that IXTn -I ::; n, also. Then let yn = X1[o,Tn) and we have which is equivalent to saying on [O,Tn ). One easily checks that (An+l)Tn - = (An)Tn -, and we can define A = An on [0, Tn), each n. The above extends without difficulty to functions 9 : ]R2 -+ ]R of the form g(Xt , H) where H is an Fo measurable random variable and x 1--4 g(x, y) is convex for every y. For general X we take Xt = X t - X o, and then j(Xt ) = j(Xt + Xo) = g(Xt , Xo), where g(x, y) = j(x + y). This completes the proof. o Notation. For x a real variable let x+, x- be the functions x+ == max(x, O) and x- == - min(x, 0). For x, y real variables, let x V y == max(x, y) and x 1\ y == min(x, y). Corollary 1. Let X be a semimartingale. Then lXI, X+, X- are all semi- martingales. 212 IV General Stochastic Integration and Local Times Proof The functions f(x) = lxi, g(x) = x+, and h(x) = x- are all convex, so the result then follows by Theorem 66. 0 Corollary 2. Let X, Y be semimartingales. Then X V Y and X 1\ Yare semimartingales. Proof Since semimartingales form a vector space and xVy = ~(Ix-YI+x+y) and x 1\ y = ~ (x + y - Ix - yl), the result is an immediate consequence of Corollary 1. 0 We can summarize the surprisingly broad stability properties of semi- martingales. Theorem 67. The space of semimartingales is a vector space, an algebra, a lattice, and is stable under C2 , and more generally under convex transforma- tions. Proof. In Chap. II we saw that semimartingales form a vector space (The- orem 1), an algebra (Corollary 2 of Theorem 22: Integration by Parts), and that they are stable under C2 transformations (Theorem 32: Ito's Formula). That they form a lattice is by Corollary 2 above, and that they are stable under convex transformations is Theorem 66. 0 Definition. The sign function is defined to be sign(x) = {I, -1, if x> 0, if x:::; o. Note that our definition of sign is not symmetric. We further define ho(x) = Ixl and ha(x) = Ix - al· Then sign(x) is the left derivative of ho(x), and sign(x-a) is the left derivative of ha(x). Since ha(x) is convex by Theorem 66 we have for a semimartingale X ha(Xt ) = IXt - al = IXo - al + t sign(Xs - - a)dXs + A~, (**) Jo+ where Af is the increasing process of Theorem 66. Using (*) and (**) as defined above we can define the local time of an arbitrary semimartingale. Definition. Let X be a semimartingale, and let ha and Aa be as defined in (*) and (**) above. The local time at a of X, denoted Lf = La(X)t, is defined to be the process given by L~ = A~ - L {ha(Xs) - ha(Xs-) - h~(Xs-)LlXs}· O 7 Local Times 213 t. Therefore so does (Afk,,:o, and finally so too does the local time Lf. We always choose this jointly measurable, cadlag version of the local time, without any special mention. We further observe that the jumps of the process Aa defined in (**) are precisely l:sa}dXs + L l{Xs _>a}(Xs - a)- 0+ O 214 IV General Stochastic Integration and Local Times Theorem 69. Let X be a semimariingale, and let Lf be its local time at the level a, each a E JR. For a.a. w, the measure in t, dLf(w), is carried by the set {s: Xs-(w) = Xs(w) = a}. Proof Since Lf has continuous paths, the measure dLf(w) is diffuse, and since {s : Xs-(w) = a} and {s : Xs-(w) = Xs(w) = a} differ by at most a countable set, it will suffice to show that dLf(w) is carried by the set {s: Xs-(w) = a}. Suppose 8, T are stopping times and that °< 8 ::; T such that [8, T) c {(s,w) : Xs-(w) < a} == {X_ < a}. Then X::; a on [8,T) as well. Hence by the first equation in Theorem 68 we have (X - a)j; - (X - a)t = iT l{X s _>a}dXs + L l{Xs _>a}(Xs - a)- S S (2) 7 Local Times 215 where X is a semimartingale and Lf = Lf(X) is its local time at a. Proof Notice that equation (*) is trivially true if f is an affine function (Le., f(x) = ax + b). Next let us first assume that J.l is a signed measure with finite total mass on a compact interval. Define a function 9 by g(x) = ~ JIx - ylJ.l(dy). It is well known that f and 9 differ by at most an affine function h(x) = ax+b. Therefore without loss of generality we can assume that the function f in (*) is of the form f(x) = ~ fix - ylJ.l(dy). Then f'(x) = ~ f sign(x - Y)J.l(dy) and f"(x) = J.l(dx). Moreover if Jl = L IXs - yl - IXs- - yl - sign(Xs _ - y)t!..Xs, O 216 IV General Stochastic Integration and Local Times inf{t > 0: IXtl ~ n}. Note that yn = I(X) on [O,Tn), and that Lrn = 0 for all a, lal ~ n, by Theorem 69. Therefore f L~J-tn(da) = f L~J-t(da) for t ::; Tn, and by the preceding (*) holds for Y = fn(X) = f(X) on [0, Tn). Therefore (*) holds on [0, Tn). Since the stopping times (Tn)n;:::l increase to ()() a.s., we have (*) holds on all of IR+ X n, and the theorem is proved. 0 The next formula gives an interpretation of semimartingale local time as an occupation density relative to the random "clock" d[X,X]~.t7 Corollary 1. Let X be a semimartingale with local time (La)aEIR. Let 9 be a bounded Borel measurable function. Then a.s.f: Lfg(a)da = it g(Xs_)d[X,X]~. Proof. Let I be convex and C2 . Comparing (*) of Theorem 70 with Ito's formula (Theorem 32 of Chap. II) shows that 100 Lff"(a)da = t f"(Xs-)d[X,X]~-00 io where J-t(da) is of course f"(a)da. Since the above holds for any continuous and positive function I", a monotone class argument shows that it must hold, up to a P-null set, for any bounded, Borel measurable function g. 0 We record here an important special case of Corollary l. Corollary 2. Let X be a semimartingale with local time (La)aEIR. Then [X,X]~ = f: Lfda. Corollary 3 (Meyer-Tanaka Formula). Let X be a semimartingale with continuous paths. Then IXtl = IXol + r t sign(Xs)dXs + L~.io+ Proof. This is merely Theorem 70 with I(x) = lxi, which implies J-t(da) 2co(da), point mass at O. The formula also follows trivially from the definition of LO. 0 If X t = Bt is a standard Brownian motion, then Mt = f; sign(Bs)dBs is a continuous local martingale, and [M, Mlt = f; sign(Bs)2d[B, B]s = [B, Blt = t. Therefore by Levy's Theorem (Theorem 39 of Chap. II) we know that M t 17 Recall that [X, Xr denotes the path-by-path continuous part of t f-> [X, Xlt with [X,X]o = O. 7 Local Times 217 is another Brownian motion. We therefore have, where Bo = x for standard Brownian motion, where f3t = f; sign(Bs)dBs is a Brownian motion, and Lt is the local time of B at zero. Formula (**) is known as Tanaka's Formula. Observe that if f(x) = lxi, then 1" = 2 218 IV General Stochastic Integration and Local Times Therefore Lt = L~ = J~ l{x.=o}dIXls' Since {Xs = O} equals {Ys = O}, this becomes Using the Meyer-Tanaka formula again we conclude that Since X is a continuous local martingale, so also is the stochastic integral on the right side above (Theorem 30); it is also non-negative, and equal to zero at O. Such a local martingale must be identically zero. This completes the prooL 0 It is worth noting that if X is a bounded, continuous martingale with X o = 0, then yt = IXtl", 0 < a < 1/2 is an example of an asymptotic martingale, or AMART, which is not a semimartingale. 18 We next wish to determine when there exists a version of the local time Lf which is jointly continuous in (a, t) f-+ Lf a.s., or jointly right continuous in (a, t) f-+ Lf. We begin with the classical result of Kolmogorov which gives a sufficient condition for joint continuity. There are several versions of Kol- mogorov's Lemma. We give here a quite general one because we will use it often in Chap. V. In this section we use only its corollary, which can also be proved directly in a fairly simple way. Before the statement of the theorem we establish some notation. Let ~ denote the dyadic rational points of the unit cube [o,l]n in JRn, and let ~m denote all x E ~ whose coordinates are of the form k2-m , 0 ~ k ~ 2m . Theorem 72 (Kolmogorov's Lemma). Let (E, d) be a complete metric space, and let UX be an E-valued random variable for all x dyadic rationals in JRn. Suppose that for all x, y, we have d(UX, UY) is a random variable and that there exist strictly positive constants E:, C, f3 such that Then for almost all w the junction x f-+ U X can be extended uniquely to a continuous junction from JRn to E. 18 For a more elementary example of an asymptotic martingale that is not a semi- martingale, see Gut [86, page 7]. 7 Local Times 219 Proof We prove the theorem for the unit cube [0, l]n and leave the extension to IRn to the reader. Two points x and y in d m are neighbors if SUPi Ixiyil = 2-m . We use Chebyshev's inequality on the inequality hypothesized to get P{d(U X,UY) ::::: Tam} ~ C2a",mTm(n+/1). Let Am = {w : :3 neighbors x, y E d m with d(UX(w), UY(w)) ::::: Tam}. Since each x E d m has at most 3n neighbors, and the cardinality of d m is 2mn , we have P(Am ) ~ c2m (a",-/1) where the constant c = 3n C. Take 0: sufficiently small so that O:c < {3. Then P(Am ) ~ cTm8 where (j = {3-o:c > o. The Borel-Cantelli Lemma then implies P(Am infinitely often) = o. That is, there exists an mo such that for m ::::: mo and every pair (u, v) of points of d m that are neighbors, We now use the preceding to show that x f-+ UX is uniformly continuous on d and hence extendable uniquely to a continuous function on [O,l]n. To this end, let x, y E d be such that Ilx - yll ~ 2- k - l . We will show that d(UX, UY) ~ c2-ak for a constant c, and this will complete the proof. Without loss of generality assume k ::::: mo. Then x = (x!, ... , xn ) and y = (yl, ... , yn) in d with Ilx - yll ~ 2-k- 1 have dyadic expansions of the form Xi = u i + 2..= a;Tj j>k yi = Vi + Lb;Tj j>k where aj, b; are each 0 or 1, and u, v are points of dk which are either equal or neighbors. Next set Uo = U, UI = Uo + ak+12-k-l, U2 = UI + ak+22-k-2, .... We also make analogous definitions for vo, VI, V2, .... Then Ui-l and Ui are equal or neighbors in dk+i, each i, and analogously for Vi-l and Vi. Hence 00 d(UX(w), UU(w)) ~ LTaj j=k 00 d(UY(w), UV(w)) ~ 2..= Taj j=k 220 IV General Stochastic Integration and Local Times and moreover The result now follows by the triangle inequality. D Comment. If the complete metric space (E, d) in Theorem 72 is separable, then the hypothesis that d(U X , UY) be measurable is satisfied. Often the metric spaces chosen are one of JR, JRd , or the function space C with the sup norm and these are separable. A complete metric space that arises often in Chap. V is the space E = v n of cadlag functions mapping [0,00) into JRn, topologized by uniform conver- gence on compacts. While this is a complete metric space, it is not separable. Indeed, a compatible metric is 00 1 dU, g) =~ 2n (1!\ o~~~n 11(8) - g(s)l). However if 1a(t) = l[a,oo)(t), then dUa, 1(3) = 1/2 for all 0:, {3 with 0 :'S: 0: < {3 :'S: 1, and since there are uncountably many such 0:, {3, the space is not separable. Fortunately, however, the condition that d(U X , UY) be measurable is nevertheless satisfied in this case, due to the path regularity of the functions in the function space V n . (Note that in many other contexts the space v n is endowed with the Skorohod topology, and with this topology V n is a complete metric space which is also separable; see for example Ethier-Kurtz [71] or Jacod-Shiryaev [110].) We state as a corollary the form of Kolmogorov's Lemma (also known as Kolmogorov's continuity criterion) that we will use in our study of local times. Corollary 1 (Kolmogorov's Continuity Criterion). Let (Xfk:~o,aEIRn be a parameterized family of stochastic processes such that t f---+ Xf is cadlag a.s., each a E JRn. Suppose that E{sup IX~ - X~IE} :'S: C(t)lla - bll n+f3 s:S,t for some c, {3 > 0, C(t) > O. Then there exists a version Xf of Xf which is B(JR+) Q9 B(JRn) Q9 :F measurable and which is cadlag in t and uniformly continuous in a on compacts and is such that for all a E JRn, t 2': 0, Xf =Xf a.s. (The null set {Xf =I- Xf} can be chosen independently of t.) In this sec- tion we will use the above corollary for parameterized processes X a which are continuous in t. In this case the process obtained from the corollary of Kolmogorov's Lemma, Xa , will be jointly continuous in (a, t) almost surely. In particular, Kolomogorov's Lemma can be used to prove that the paths of standard Brownian motion are continuous. 7 Local Times 221 Corollary 2. Let B be standard Brownian motion. Then there is a version of B with continuous paths, a.s. Proof Since Bt - B s is Gaussian with mean zero and variance t - s, we know that E{IBt - B s 14} ~ c(t - s)2. (One can give a cute proof of this moment estimate using the scaling property of Brownian motion.) If we think of time as the parameter and the process as being constant in time, we see that the exponent 4 is strictly positive, and that the exponent on the right, 2, is strictly bigger than the dimension, which is of course 1. Corollary 2 now follows from Corollary 1. 0 Hypothesis A. For the remainder of this section we let X denote a semi- martingale with the restriction that L:o O. Observe that if (n, F, (Ft)t>o, P) is a probability space where (Ft)t>o is the completed minimal filtratiO"n of a Brownian motion B = (Bdt>o, then all semimartingales on this Brownian space verify Hypothesis A. Indeed, by Corollary 1 of Theorem 43 all the local martingales are continuous. Thus if X is a semimartingale, let X = M + A be a decomposition with M a local martingale and A an FV process. Then the jump processes LlX and LlA are equal, hence L ILlXsl = L ILlAsl < 00, O 222 IV General Stochastic Integration and Local Times This notation should not be confused with that of Kolmogorov's Lemma (The- orem 72). It is always assumed that we take the B(IR) 18iB(IR+) 18iF measurable, cadlag version of za (d., Theorem 63). Before we can prove our primary regularity property (Theorem 75), we need two preliminary results. The first is a very special case of a family of martingale inequalities known as the Burkholder-Davis-Gundy inequalities. Theorem 73 (Burkholder's Inequality). Let X be a continuous local mar- tingale with X o = 0, 2 ~ p < 00, and T a finite stopping time. Then E{(XT)P} ~ CpE{[X, X]!f'2} where Cp = {qP(P(P;1»}P/2, with ~ + ~ = 1. Proof By stopping, it suffices to consider the case where X and [X, X] are bounded. By Ito's formula we have IXTIP = p rT sign(Xs)IXsIP-1dXs + p(p - 1) r T IXsIP-2d[X, XJs.Jo 2 Jo By Doob's inequalities (Theorem 20 of Chap. I) we have (with ~ + ~ = 1) E{(XT)P} ~ qPE{IXTIP} = qPE{P(P; 1) iT IXsIP-2d[X,X]s} ~ qP (P(P; 1») E{(XT )P-2[X,X]r} ~ qP (P(P; 1») E{(XT )P}1-%E{[X,X]!f'2}2/P, with the last inequality by Holder's inequality. Since E{(X;)p}l-% < 00, we divide both sides by it to obtain and raising both sides to the power p/2 gives the result. D Actually much more is true. Indeed for any local martingale (continuous or not) it is known that there exist constants cp , Cp such that for a finite stopping time T E{(XT)P}l/P ~ cpE{[X, X]!f'2}1/P ~ CpE{(XT)P}l/p for 1 ~ P < 00. See Sect. 3 of Chap. VII of Dellacherie-Meyer [46] for these and related results. When the local martingales are continuous some results even hold for 0 < P < 1 (see, e.g., Barlow-Jacka-Yor [9, Table 4.1 page 162]). 7 Local Times 223 Theorem 74. Let X be a semimartingale satisfying Hypothesis A. There ex- ists a version of (XC)f such that (a,t,w) f---+ (XC)f(w) is B(IR)I8iP measurable, and everywhere jointly continuous in (a, t). Proof Without loss of generality we may assume X - Xo E '}-{2. If it is not, we can stop X - Xo at T n -. The continuous local martingale part of XT n - is then just (xc)Tn • Suppose -00 < a < b < 00, and let cxt(a,b) = E{(l t 1{b~xs_>a}d[X,X]~)2}. By Corollary 1 of Theorem 70 we have cxt(a, b) = E{(l b Lfdu)2} = (b - a)2E{(b ~ alb Lfdu)2} ~ (b-a)2E{b~al b(Lf)2 du }, by the Cauchy-Schwarz inequality. The above implies CXt(a, b) ~ (b - a)2 sup E{(Lf)2}. uE(a,b) By the definition, Lf ~ Af ~ IXt - Xol - {t sign(Xs _ - u)dXs . io+ and therefore E{(Lf)2} ~ 2E{IXt - X OI2 } + 2E{( {t sign(Xs _ - u)dXs )2}io+ ~ 411X - Xoll~2 + 411 sign(Xs _ - u) . (X - Xo)II~2 ~ 811X - Xoll~2 < 00, and the bound is independent of u. Therefore for a constant r < 00, and independent of t. Next using Burkholder's inequal- ity (Theorem 73) we have E{sup I(XC)~ - (xc)~14} ~ C4 E{( {= 1{b~xs_>a}d[X,X]~)2} s io ~ C4 supcxt(a, b) t ~ C4r(b - a)2. The result now follows by applying Kolmogorov's Lemma (the corollary of Theorem 72). 0 224 IV General Stochastic Integration and Local Times We can now establish our primary result. Theorem 75. Let X be a semimartingale satisfying Hypothesis A. Then there exists a B(lR) 18iP measurable version of (a, t, w) f-+ L~(w) which is everywhere jointly right continuous in a and continuous in t. Moreover a.s. the limits La- l' Lb' tt = Imb-->a, b 7 Local Times 225 An example of a semimartingale satisfying Hypothesis A but having a discontinuous local time is X t = IBtl where B is standard Brownian motion with Bo = o. Here La(x) = La(B) + L -a(B) for a > 0, LO(X) = LO(B), and La(X) = 0 for a < o. J. Walsh [227] has given a more interesting (but more complicated) example. Corollary 2. Let X be a semimartingale satisfying Hypothesis A. Let A be as in Corollary 1. The local time (L't) is continuous in t and is continuous at a = ao if and only if 100 l{Xs =ao}ldAs l = o. Observe that if X = B, a Brownian motion (or any continuous local mar- tingale), then A = 0 and the local time of X can be taken everywhere jointly continuous. Corollary 3. Let X be a semimartingale satisfying Hypothesis A. Then for every (a, t) we have and 1ltL~- = lim - l{a-E 0, x :S O. A symmetrized result follows trivially, namely La+La- 1 lt t t = lim - l{IX -alO 2€ ° s_ a.s. Corollary 3 is intuitively very appealing, and justifies thinking of local time as an occupation time density. Also if X = B, a Brownian motion, then Corollary 3 becomes the classical result 1 ltL~ = lim - l(a_E,a+E)(Bs )ds, E->O 2€ ° a.s. Local times have interesting properties when viewed as processes with "time" fixed, and the space variable "a" as the parameter. The Ray-Knight Theorem giving the distribution of (L}-a)o:$a 226 IV General Stochastic Integration and Local Times The Meyer-Ito formula (Theorem 70) can be extended in a different di- rection, which gives rise to the Bouleau-Yor formula, allowing non-convex functions of semimartingales. The key idea is that the function a f-+ Lt in- duces a measure on JR if L is the local time of a semimartingale X satisfying Hypothesis A and U is a positive random variable. Theorem 76. Let X be a semimartingale satisfying Hypothesis A, U a pos- itive mndom variable, La the local times of X. Then the opemtion n f f-+ ~ fi(L'tr' - L'tJ), i=l where f (x) = L fi 1(ai ,ai+ ,] (x), can be extended uniquely to a vector measure on B(IR) with values in LO. Proof. Recall that LO denotes finite-valued random variables. By Theorem 68 we know that We write Sf for the last two sums on the right side of the equation above. Note first that (Xu - a)- - (Xo - a)- is Lipschitz continuous in a and hence absolutely continuous in a. Therefore it induces a measure in a. Also the function a f-+ Su is cadlag and of finite variation in a, and moreover f IdaSul ~ 2 I:o 8 Azema's Martingale 227 Corollary. Let X be a semimartingale satisfying Hypothesis A, U a positive random variable, and Xf = J~ l{Xs _::Oa}dXs. Then daXucan be defined as an LO-valued measure, and for f E bB(lR.) we have Combining Theorem 76 and its corollary yields the Bouleau-Yor formula. We leave its proof to the reader. Theorem 77 (Bouleau-Yor Formula). Let X be a semimartingale satisfy- ing Hypothesis A, U a positive random variable, f a bounded, Borel function, and F(x) = J: f(u)du. Then F(Xu ) - F(Xo) = (u f(Xs-)dXs - ~ jf(a)daLuJo+ 2 + L {F(Xs) - F(Xs_) - f(Xs-)llXs}' o 228 IV General Stochastic Integration and Local Times sign(x) = {I, -1, if x> 0, if x:::; 0. Set 9~ = O"{sign(Bs ); s :::; t}, and let O. Then P(sup B s > c) = 2P(Bt > c). s::O;t An elementary conditioning argument provides a useful corollary. Corollary. Let B be standard Brownian motion, Bo = 0, and 0 < s < t. Then P( sup Bu > O,Bs < 0) = 2P(Bt > O,Bs < 0). s 8 Azema's Martingale 229 P(9t > s) = 2P(9t > s, B s < 0) = 2P( sup B u > O,Bs < 0) s s : Bu = O}. Note that T is a bounded stopping time. Then 230 IV General Stochastic Integration and Local Times E{IBtligt::; s} = 2E{Bt i9t::; s and B s > O} = 2E{BtiT = t and B s > O} = 2E{BTiBs > O} = 2E{Bs;B s > O}, since BT = 0 on {T of t}, and the last step uses that B is a martingale, and s ::; T. The last term above equals E{IBsl} = {fy's. Therefore E{IB II = s} = fgE{IBtl;gt::; s} t gt Jip( < s)ds gt_ =Ii~, since f P(gt ::; s) = ~, by Theorem 79. s 71" s(t-s) o Let 1f? = O"{Msi s ::; t}, the minimal filtration of Azema's martingale M, and let JH[ = (1ftk~o be the completed filtration. Corollary. The two filtrations JH[ and 8 Azema's Martingale 231 Proof By Theorem 69 we know that the measure dL~(w) on lR.+ is carried by the set {s: Ms-(w) = Ms(w) = a}, a.s., for each a. However if a of 0, this set is countable. Since s f--+ L~ is a.s. continuous, it cannot charge a countable set. Therefore dL~ is the zero measure for a of O. Since Lg = 0, we have Lf := 0 for a of O. 0 Theorems 81 and 82 together show that Azema's martingale provides a counterexample to a general version of Theorem 75, for example. Therefore a hypothesis such as Hypothesis A is necessary, and also we see that Azema's martingale does not satisfy Hypothesis A. Since all semimartingales for the Brownian motion filtration IF do satisfy Hypothesis A, as noted in Sect. 7, we conclude the following. Theorem 83. Azema's martingale M is not a semimartingale for the Brown- ian filtration IF. While the local time at zero, LO(M), is non-trivial by Theorem 81, more information is available. It is actually the same as the Brownian local time, as Theorem 85 below shows. First we need a preliminary result. Theorem 84. The local time LO(B) of Brownian motion is 232 IV General Stochastic Integration and Local Times Corollary 1. The process IMI - LO(B) is a G martingale. Proof This corollary is proved at the end of the proof of Theorem 85. 0 Corollary 2. The process l{Bt>O}ylIVt - gt - !L~(B) is a o}M; = l{M s _:'Oo}M; = 0, it follows from Theorem 68 that M/ - ~L~U\,f) is a martingale. However M t+ = l{Bt>O}~Vt - gt by Theorem 80, and L~(M) = L~(B) by Theorem 85, and the theorem follows. o The next theorem shows that Azema's martingale is quadratic pure jump. Theorem 86. For Azema's martingale M, [M,M]C == 0, and [M,M]t = ~gt. Proof By Corollary 2 of Theorem 70, [M,MJf = I: Lfda. By Theorem 82, [M,M]C == O. We conclude that [M,M]t = LO 9 Sigma Martingales 233 9 Sigma Martingales We saw with Emery's example (which precedes Theorem 34) a stochastic integral with respect to an fi2 martingale that was not even a local martingale. Yet "morally" we feel that it should be one. It is therefore interesting to consider the space of stochastic integrals with respect to local martingales. We know these are of course semimartingales, but we want to characterize the class of processes that arise in a more specific yet reasonable way. By analogy, recall that in measure theory a a-finite measure is a measure p, on a space (8,8) such that there exists a sequence of measurable sets Ai E 8 such that Ui Ai = 8 and p,(Ai) < 00, each i. Such a measure can be thought of as a special case of a countable sum of probability measures. It is essentially a similar phenomenon that brings us out of the class of local martingales. Definition. An ll~d-valuedsemimartingale X is called a sigma martingale if there exists an IRd-valued martingale M and a predictable 1R+-valued process H E L(M) such that X = H . M. In the next theorem, H will always denote a strictly positive predictable process. Theorem 88. Let X be a semimartingale. The following are equivalent: (i) X is a sigma martingale; (ii) X = H . M where M is a local martingale; (iii) X = H . M where M is a martingale; (iv) X = H . M where M is a martingale in fil . Proof It is clear that (iv) implies (iii), that (iii) implies (ii), and that (iii) obviously implies (i), so we need to prove only that (i) implies (iv). Without loss assume that M o = O. Since X is a sigma martingale it has a representation of the form X = H . M where M is a martingale. We know that we can localize M in fil by Theorem 51, so let Tn be a sequence of stopping times tending to 00 a.s. such that MTn E fii for each n. Set To = 0 and let Nn = l(Tn_t ,Tn] . MTn. Let an be a strictly positive sequence of real numbers such that L n anllNnllw < 00. Define N = L n anNn and one can check that N is an fii martingale. Set J = l{H=O} +HLn a;;:I1(Tn_t,TnJ. Then it is simple to check that X = J . N and that J is strictly positive. 0 Corollary 1. A local sigma martingale is a sigma martingale. Proof Let X be a local sigma martingale, so that there exists a sequence of stopping times tending to 00 a.s. such that XTn is a sigma martingale for each n. Since XTn is a sigma martingale, there exists a martingale M n in fil such that XTn = Hn . Mn, for each n, where Hn is positive predictable. Choose ¢n positive, predictable so that II¢nHn. MTnllw < Tn. Set To = 0 and ¢o = ¢I1{o}. Then ¢ = ¢o + Ln>I ¢n1(Tn_t,Tn] is strictly positive and predictable. Moreover ¢ . X is an fii ~artingale,whence X = i .¢ . X, and the corollary is established. 0 234 IV General Stochastic Integration and Local Times Corollary 2. A local martingale is a sigma martingale. Proof This is simply a consequence of the fact that a local martingale is locally a martingale, and trivially a martingale is a sigma martingale. 0 The next theorem gives a satisfying stability result, showing that sigma martingales are stable with respect to stochastic integration. Theorem 89. If X is a local martingale and H E L(X), then the stochastic integral H . X is a sigma martingale. Moreover if X is a sigma martingale (and a fortiori a semimartingale) and H E L(X), then H . X is a sigma martingale. Proof Clearly it suffices to prove the second statement, since a local mar- tingale is already a sigma martingale. But the second statement is simple. Since X is a sigma martingale we know an equivalent condition is that there exists a strictly positive predictable process ¢ and a local martingale M such that X = ¢ . M. Since H is predictable, the processes HI = Hl{H>O} + 1 and H2 = -Hl{H 0 everywhere, such that H . X is a martingale, which we can easily do using Theorem 88. Then (H 1\ 1)· X = (Hfil). (H. X) is a local martingale, and hence without loss of generality we can assume H is bounded. Then H . A is a finite variation predictable process, and we have H . A = H . X - H . M which is a local martingale. Thus we conclude that H . A = O. If we replace H with JH where J is predictable, IJI :::; 1, and is such that J~ JsdAs = J~ IdAsI, then we conclude that J~ HsIdAs1= 0, and since H > 0 everywhere, we conclude that A = 0 establishing the result. 0 We now have that any criterion that ensures that the semimartingale X is special, will also imply that the sigma martingale X is a local martingale. Bibliographic Notes 235 Corollary 1. If a sigma martingale X has continuous paths, then it is a local martingale. Corollary 2. If X is a sigma martingale and if either X; = sUPs 236 Exercises for Chapter IV efforts of many researchers, culminating in Meyer's paper [170J; we follow the treatment in [87J. The Jacod-Yor Theorem is from [111], and Sect. 9 largely follows Emery [69]. The theory of stochastic integration depending on a parameter is due to the fundamental work of Stricker-Yor [219], who used a key idea of DoIeans- Dade [48]. The Fubini Theorems for stochastic integration have their origins in the book of Doob [55] and Kallianpur-Striebel [120] for the Ito integral, Kailath-Segall-Zakai [119J for martingale integrals, and Jacod [103] for semi- martingales. The counterexample to a general Fubini Theorem presented here is due to Janson. The theory of semimartingale local time is of course abstracted from Brow- nian local time, which is due to Levy [149]. It was related to stochastic inte- gration by Tanaka (see McKean [153]) for Brownian motion and the Ito inte- gral. The theory presented here for semimartingale local time is due largely to Meyer [171]. See also Millar [181J. The measure theory needed for the rigorous development of semimartingale local time was developed by Stricker-Yor [219], and the Meyer-Ito formula (Theorem 70) was formally presented in Yor [242], as well as in Jacod [103]. Theorem 71 is due to Yor [244], and has been ex- tended by Ouknine [188]. A more general version (but more complicated) is in Qinlar-Jacod-Protter-Sharpe [34]. The proof of Kolmogorov's Lemma given here is from Meyer [177]. The results proved under Hypothesis A are all due to Yor [243J except the Bouleau-Yor formula [22]. See Bouleau [21] for more on this formula. Azema's martingale is of course due to Azema [4], though our presenta- tion of it is new and it is due largely to Janson. For many more interesting results concerning Azema's martingale, see Azema-Yor [5], Emery [69], and Meyer [179]. Sigma martingales date back to the work of Emery [67], and to Chou [29]. The importance of sigma martingales was clarified through the work in math- ematical finance of Delbaen and Schachermayer [38J. See [110J for a more comprehensive treatment. Exercises for Chapter IV Exercise 1. Let X be a semimartingale. Show that X is special if and only if the increasing process C t = [X, X]t is locally integrable; that is, there exists a sequence of stopping times Tn increasing to infinity a.s. such that E{[X,X]Tn } < 00 for each time Tn- *Exercise 2. Let X be a special semimartingale with canonical decomposi- tion X t = X o + Mt + At. Show that the following two inequalities hold: E{[A,A]oo}::; E{[X,XJoo} and E{[M,M]oo}::; 4E{[X,X]oo}' Exercises for Chapter IV 237 Exercise 3. Let (D, F, P) be a complete probability space. Let X n be a sequence of random variables such that P(lXnl < 00) = 1, all n. Show that there exists another probability Q which is equivalent to P in the sense that the null sets are the same for both measures, with 'f!i, bounded, and every random variable X n is in Ll(dQ). Exercise 4. Let X be a semimartingale. Show that there is a probability measure Q, equivalent to P, such that under Q, X has a decomposition X = M + A, where M is a martingale with EQ([M, M]t) < 00 for each t, 0 ::; t < 00, and A is a predictable process of finite variation on compacts with EQU; IdAsl) < 00 for each t, 0::; t < 00. (Hint: Use Exercises 1, 2, and 3.) *Exercise 5. Let xn be a sequence of semimartingales. Show that there exists one probability measure Q, equivalent to P, such that under Q, each xn has a decomposition xn = Mn + An, where Mn is a martingale with EQ([Mn, Mn]t) < 00 for each t, 0::; t < 00 and each n, and An is a predictable process of finite variation on compacts with EQU; IdA~1) < 00 for each t, o ::; t < 00 and each n. Exercise 6. Let S denote the space of all square integrable martingales with continuous paths a.s., on a given filtered complete probability space satisfying the usual conditions. Show that S is a stable subspace. (This exercise was referred to in Definition 3.) Exercise 7. Let Z be a Levy process with E{IZtl} < 00 and E{Zt} = 0 for all t ~ O. Show that Z is a martingale, and that ZC is either 0 or Brownian motion, where ZC is defined in Exercise 6. Conclude that any martingale Levy process Z that has no Brownian component is purely discontinuous. Exercise 8. Let N be a standard Poisson process with arrival intensity A. Show that the martingale Nt - At is purely discontinuous. (Note that this gives an example of a purely discontinuous martingale whose sample paths are not purely discontinuous in the sense that they do not change only by jumps.) Exercise 9. Let M be a square integrable martingale. Show that [M, M]f = [MC, MC]t = [MC, M]t, all t ~ 0, almost surely. Exercise 10 (example of orthogonal projection). Let T be a stopping time. Show that the collection S of all square integrable martingales stopped at T is a stable subspace. Also show that if M is an arbitrary square integrable martingale, then its orthogonal decomposition with respect to this subspace is M = MT + (M - MT). Give a description of the orthocomplement of S. Exercise 11 (example of orthogonal projection). Let T be a totally inaccessible stopping time and let M be a square integrable martingale. Let Ut = .6.MT 1{t2:T } and let L be the martingale of U minus its compensator. We denote L by IIM. Show that IIM and M - IIM are orthogonal. Let S denote the space of all square integrable martingales which are continuous at the instant T. Show that S is a stable subspace, and that IIM represents the orthogonal projection of M onto the orthocomplement of S. 238 Exercises for Chapter IV *Exercise 12. Let T be an arbitrary stopping time, T > 0, and let X be a random variable in L 1 . Show that M = X1{t2T} is a uniformly integrable martingale if and only if E{XIFT _} = 0. *Exercise 13. Let T be a stopping time, T > 0, and let M be a square integrable martingale. Show that the decomposition is an orthogonal decomposition. Show that one of the stable subspaces in question is the space of all square integrable martingales M such that MT is FT- measurable. (Hint: Use Exercise 12.) *Exercise 14. Let Z be a Levy process on a complete probability space (D., F, P) and let IF be its minimal filtration completed (and satisfying the usual hypotheses). Show that (D., F, IF, P) is an absolutely continuous space. Exercise 15. Consider the structure equation d[X, X]t = dt + f3Xt_dXt . Show that this equation has a scaling property: if X is a martingale solu- tion, then so too is t X.\2t for every>. 1:- 0. Note that by the uniqueness in distribution of the solution, the processes (Xdt20 and (tX.\2t)t20 have the same distribution. *Exercise 16. Let X be a martingale solution of the structure equation of Exercise 15. Show that XC = 0, where XC is defined in Exercise 6. Exercise 17. Let M be a local martingale in 1{P for some P 2 1. Show that M is a uniformly integrable martingale. Show further that if p = 1 then E{Moo } :S CIIMII7-(l where the constant c is universal. That is, c does not depend on M. Exercise 18. Suppose that M is a bounded martingale. Show that This can improved to replace v5 with 2. Exercise 19. Show that there are martingales with finite BMO norm which are not bounded. (Hint: Let M be a bounded martingale and H be a pre- dictable process such that IHI :S 1, and let N = H . M. Then it is easy to see that N is in BMO, and to construct such an N that is not bounded.) Exercise 20. Prove this version of the Fundamental Theorem of Local Mar- tingales. Let M be a uniformly integrable martingale. Show that there exist N and U such that N is in B M 0, U is of integrable variation, and M = U + V. If M is a local martingale this holds locally. Exercise 21. Show that an arbitrary local martingale is a semimartingale by using the Burkholder-Davis-Gundy inequalities instead of the Fundamental Theorem of Local Martingales. (See [61].) Exercises for Chapter IV 239 Exercise 22. Let M be a local martingale. IfT is a predictable stopping time, then 1[0,T) (s) E bP so that Mr- = I; 1[0,T) (s )dMs is a local martingale. Give an example of a local martingale M and a stopping time T such that M T - is not a local martingale. Exercise 23. Let X be a continuous semimartingale. Show that [lXI, X] = [X, X]. (Hint: Use local times.) *Exercise 24. Let B be standard Brownian motion and let 0: > 0. Show that 101 IBtl-adt < 00 a.s. if and only if 0: < 1. (Hint: Use local times.) Exercise 25. Let X be a semimartingale. Show that L~(X) = ~L~(X+) where X+ = max(X, 0) and L~(X) denotes the local time of X at time t and level 0. Exercise 26. Let A be adapted, continuous, and of finite variation on com- pacts, and let B be standard Brownian motion. Let A denote Lebesgue mea- sure on the line. Show that A(S : B s = As) = °a.s. Exercise 27. Let X be a continuous semimartingale. Show that L~ E FJxl, each t ;::: 0, where FJxl denotes the smallest right continuous completed fil- tration generated by the process (IXtlk~o. Exercise 28. Let X be a semimartingale. Show that for each fixed (w, t) the section x J---> Lt(X) has compact support. Exercise 29 (Skorohod's Lemma). This result will be useful for the next two exercises. Let x be a real valued continuous function on [0, 00) such that x(o) ;::: 0. Show there exists a pair of functions (y,a) on [0,00) such that x = y + a, y is positive, and a is increasing, continuous, a(O) = 0, and the measure das induced by a is carried by the set {s : y(s) = o}. Show further that the function a is given by a(t) = sUPs~t( -x(s) V 0). Exercise 30. Let M be a continuous local martingale with Mo = 0. Let IM t I = Nt + L~ (M). Show that N is a continuous local martingale with No = 0, and that L~(M) = sUPs9(-Ns ). *Exercise 31. Let X be a continuous semimartingale, and suppose Y satisfies the equation dyt = - sign(yt)dXt · Show that Iytl = xt - X t , where Xt = sUPu~t Xu' (Hint: Use Tanaka's formula and Exercise 29.) Exercise 32. Suppose that X and Yare continuous semimartingales such that L~(Y - X) = 0. Show that L~(X V Y) = it 1{Y8~0}dL~(X)+ it l{Xso}dL~(X)+ it l{Xs~o}dL~(Y)+ it l{Xs=Ys=o}dL~(Y+- X+). 240 Exercises for Chapter IV Exercise 34 (Ouknine's Formula). Let X and Y be continuous semi- martingales. Show that L~(X V Y) + L~(X 1\ Y) = L~(X) + L~(Y). *Exercise 35 (Emery-Perkins Theorem). Let B be standard Brownian motion and set L t = L~(B). Let X = B + eL, e E lR. fixed. Assume that there exists a set D C lR.+ x D such that (i) D is predictable for F X ; (ii) P(w: (t,w) E D) = 1 each t > 0; and (iii) P(w : (Tt(w),w) ED) = °each t> 0, where Tt = inf(s > °:L s > t). (a) Show that B t = J~ ID(S)dXs ' Also show that B t E F X , each t > 0, and conclude that F B = F X = F B+cL , all e E R (b) Let D ={(t,W): lim ~ l{x _k(W)-X -(k-l) (w»O} =~}. n-->oo L....t t-2 t-2 2 k=l Use the fact that s J---> (BSI\T" LSI\Tt ) and s J---> (-BTt_s,LTt -LTt_J have the same joint distribution to show that D satisfies (i), (ii), and (iii). *Exercise 36. With the notation of Exercise 35, let Y = IBI +eL. Show that F Y c FIBI, for all e. *Exercise 37. With the notation of Exercise 36, Pitman's Theorem states that Y is a Bessel process of order 3, which in turn implies that for F Y , with e = 1, t 1 yt = (3t + io Y s ds for an F Y-Brownian motion (3. Use this to show that FY 1:- FIBI for e = 1. (Note: One can show that F Y = FIBI for all e =1= 1.) *Exercise 38 (Barlow's Theorem). Let X solve the equation where B is a standard Brownian motion. The above equation is said to have a unique weak solution if whenever X and Yare two solutions, and if !Lx is defined by !Lx(A) = P{w : t J---> Xt(w) E A}, then !Lx = !LY. The above equation is said to have a unique strong solution if whenever both X and Yare two solutions on the probability space on which the Brownian motion B is defined, then P{w : t J---> Xt(w) = t J---> yt(w)} = 1. Assume that the equation has a unique weak solution with the property that if X, Yare any two solutions defined on the same probability space, then for all t :::: 0, L~(X - Y) = 0. Show that the equation has a unique strong solution. Exercises for Chapter IV 241 Exercise 39. Consider the situation of Exercise 38, and suppose that u and b are bounded Borel. Further, suppose that (u(x) - u(y))2 ::; p(lx - yl) for all x, y, where p: [0,00) -+ [0,00) is increasing with J;+ p(~) du = +00 for every c > 0. Show that if X, Yare two solutions, then L~(X - Y) = O. *Exercise 40. Let B be a standard Brownian motion and consider the equa- tion Xt= it sign(Xs)dBs' Show that the equation has a unique weak solution, but that if X is a solution then so too is -X. (See Exercise 38 for a definition of weak and strong solu- tions.) Hence, the equation does not have a unique strong solution. Finally, show that if X is a solution, then X is not adapted to the filtration:FE . (Hint: Show first that B t = J; l{IX s l#O}dIXsl·) Exercise 41. Let X be a sigma martingale and suppose X is bounded below. Show that X is in fact a local martingale. *Exercise 42. Show that any Levy process which is a sigma martingale is actually a martingale (and not just a local martingale). Exercise 43. Let (Un)n::~l be i.i.d. random variables with P(Ul = 1) = P(Ul = -1) = 1/2, and let X = En>l 2-nUn1{t~qn} where (qn)n~l is an enumeration of the rationals in (0,1). Let Ht = En>l ~ l{t~qn} and show that X E 1{2 and H E L(X), but also that X is ~f finite variation and Y = H . X has infinite variation. Thus the space of finite variation processes is not closed under stochastic integration! *Exercise 44. Let xn be a sequence of semimartingales on a filtered complete probability space (D,:F, IF, P) satisfying the usual hypotheses. Show there ex- ists one probability Q, which is equivalent to P, with a bounded density such that under Q each xn is a semimartingale in 1{2. Exercise 45. Let B be a standard Brownian motion and let LX be its local time at the level x. Let A = {w: dL~(w) is singular as a measure with respect to ds} where of course ds denotes Lebesgue measure on lR.+. Show that P(A) = 1. Conclude that almost surely the paths s f---+ L'; are not absolutely continuous. (Hint: Use Theorem 69.) Exercise 46. Let B be a standard Brownian motion and let N be a Poisson process with parameter>. = 1, with B and N independent as processes. Let L be the local time process for B at level 0, and let X be the vector process X t = (Bt , N L,). Let IF be the minimal filtration of X, completed in the usual way. (a) Show that X is a strong Markov process. 242 Exercises for Chapter IV (b) Let T be the first jump time of X. Show that the compensator of the process Ct = l{t~T} is the process At = LtIlT . Note that this gives an example of a compensator of the first jump time of a strong Markov process which has paths that are almost surely not absolutely continuous. **Exercise 47. Let Z be a Levy process. Show that if Z is a sigma martingale, then Z is a martingale. (Note: This exercise is closely related to Exercise 29 of Chap. I.) v Stochastic Differential Equations 1 Introduction A diffusion can be thought of as a strong Markov process (in JRn) with contin- uous paths. Before the development of Ito's theory of stochastic integration for Brownian motion, the primary method of studying diffusions was to study their transition semigroups. This was equivalent to studying the infinitesimal generators of their semigroups , which are partial differential operators. Thus Feller's investigations of diffusions (for example) were actually investigations of partial differential equations, inspired by diffusions. The primary tool to study diffusions was Kolmogorov's differential equa- tions and Feller's extensions of them. Such approaches did not permit an analysis of the paths of diffusions and their properties. Inspired by Levy's in- vestigations of sample paths, ItO studied diffusions that could be represented as solutions of stochastic differential equations l of the form where B is a Brownian motion in JRn , u is an n x n matrix, and b is an n-vector of appropriately smooth functions to ensure the existence and uniqueness of solutions. This gives immediate intuitive meaning. If (Ftk:~o is the underlying filtration for the Brownian motion B, then for small c > 0 E{Xf+c - XfIFt} = bi(Xt)c + o(c) E{(Xf+c - Xl - cbi(Xt))(Xl+c - xl- c[r1(Xt ))IFt} = (uu')ij(Xt)c + o(c), where u' denotes the transpose of the matrix u. ItO's differential "dB" was found to have other interpretations as well. In particular, "dB" can be thought of as "white noise" in statistical communi- cation theory. Thus if ~t is white noise at time t, B t = J; ~sds, an equation 1 More properly (but less often) called "stochastic integral equations." 244 V Stochastic Differential Equations which can be given a rigorous meaning using the theory of generalized func- tions (cf., e.g., Arnold [2]). Here the Markov nature of the solutions is not as important, and coefficients that are functionals of the paths of the solutions can be considered. Finally it is now possible to consider semimartingale driving terms (or "semimartingale noise"), and to study stochastic differential equations in full generality. Since "dB" and "dt" are semimartingale differentials, they are always included in our results as special cases. While our treatment is very general, it is not always the most general available. We have at times preferred to keep proofs simple and non-technical rather than to achieve maximum generality. The study of stochastic differential equations (SDEs) driven by general semimartingales (rather than just by dB, dt, dN, and combinations thereof, where N is a Poisson process) allows one to see which properties of the so- lutions are due to certain special properties of Brownian motion, and which are true in general. For example, in Sect. 6 we see that the Markov nature of the solutions is due to the independence of the increments of the differentials. In Sects. 8 and 10 we see precisely how the homeomorphic and diffeomorphic nature of the flow of the solution is a consequence of path continuity. In Sect. 5 we study Fisk-Stratonovich equations which reveal that the "correction" term is due to the continuous part of the quadratic variation of the differentials. In Sect. 11 we illustrate when standard moment estimates on solutions of SDEs driven by Brownian motion and dt can be extended to solutions of SDEs driven by Levy processes. 2 The HP Norms for Semimartingales We defined an 1{2 norm for semimartingales in Chap. IV as follows. If X is a special semimartingale with X o = 0 and canonical decomposition X = N +A, then We now use an equivalent norm. To avoid confusion, we write H 2 instead of 1{2. Moreover we will define HP, 1 :::; p :::; 00. We begin, however, with a different norm on the space Jl)l (i.e., the space of adapted cadlag processes). For a process H E Jl)l we define H* = sup IHtl, t IIHllsp = IIH*IILP. Occasionally if H is in lL (adapted and caglad) we write IIHII~p as well, where the meaning is clear. 2 The HP Norms for Semimartingales 245 If A is a semimartingale with paths of finite variation, a natural definition of a norm would be IIAllp = II Jooo IdAsIIILP, where IdAs(w)1 denotes the total variation measure on JR.+ induced by s J---> As (w). Since semimartingales do not in general have such nice paths, however, such a norm is not appropriate. Throughout this chapter, we will let Z denote a semimartingale with Zo = 0, a.s. Let Z be an arbitrary semimartingale (with Zo = 0). By the Bichteler- Dellacherie Theorem (Theorem 43 of Chap. III) we know there exists at least one decomposition Z = N + A, with N a local martingale and A an adapted, cadlag process, with paths of finite variation (also No = Ao = 0 a.s.). For 1 :::; p :::; 00 we set Definition. Let Z be a semimartingale. For 1 :::; p :::; 00 define IIZIIHP = inf jp(N, A) = Z=N+A where the infimum is taken over all possible decompositions Z = N +A where N is a local martingale, A E j[]) with paths of finite variation on compacts, and Ao = No = O. The corollary of Theorem 1 below shows that this norm generalizes the HP norm for local martingales, which has given rise to a martingale theory analogous to the theory of Hardy spaces in complex analysis. We do not pursue this topic (d., e.g., Dellacherie-Meyer [46]). Theorem 1. Let Z be a semimartingale (Zo = 0). Then II[Z, Z]~21ILP < IIZIIHP, (1 ::; p::; 00). Proof Let Z = M + A, Mo = Ao = 0, be a decomposition of Z. Then [Z, Z]~2 :::; [M, MJ~2 + [A, A];S2 = [M,M]~2 + (L(L~As)2)l/2 :::; [M, Ml~2 +L I.6.Asl s where the equality above holds because A is a quadratic pure jump semi- martingale. Taking LP norms yields II[Z,Z]~21ILP :::; jp(M,A) and the result follows. D Corollary. If Z is a local martingale (Zo = 0), then IIZIII1P = II[Z, ZJ;S2 1ILP , 246 V Stochastic Differential Equations Proof. Since Z is a local martingale, we have that Z = Z +0 is a decomposition of Z. Therefore IIZIIHP :::: jp(Z, 0) = II[Z, ZJ~,:?lb· By Theorem 1 we have II[Z, Z]ZxS2 1ILP :::: IIZllltt' hence we have equality. 0 Theorem 2 is analogous to Theorem 5 of Chap. IV. For most of the proofs which follow we need only the case p = 2. Since this case does not need Burkholder's inequalities, we distinguish it from the other cases in the proof. Theorem 2. For 1 :::: p < 00 there exists a constant cp such that for any semimartingale Z, Zo = 0, II ZIIg:p :::: cpll ZIIMP' Proof A semimartingale Z is in ]j)), so IIZII~p makes sense. Let Z = M + A be a decomposition with Mo = Ao = O. Then IIZII£p = E{(Z~)P} :::: E{(M~ + lOO'dAs')P} :::: CpE{(M~)P + ~ooldAsI)P}, using (a + b)P :::: 2P- 1(aP+ bP). In the case p = 2 we have by Doob's maximal quadratic inequality that E{(M~)2} :::: 4E{M~J = 4E{[M, M]oo}. For general p, 1 :::: p < 00, we need Burkholder's inequalities, which state for a universal constant cp which depends only on p and not on the local martingale M.For continuous local martingales and p 2 2 we proved this using ItO's formula in Chap. IV (Theorem 73). For general local martingales and for all finite p 2 1 see, for example, Dellacherie-Meyer [46, page 287J. Continuing, letting the constant cp vary from line to line we have IIZII£p :::: cpE{(M~,)P + (100 IdAsl)P} :::: cpE{[M, MJ:2 + (100 IdAsI)P} :::: cp[jp(M,A)]p, and taking p-th roots yields the result. o Corollary. On the space of semimartingales, the HP norm is stronger than the gP norm, 1:::: p < 00. - Theorem 3 (Emery's Inequality). Let Z be a semimariingale, H E IL, and ~ + %= ~ (1 :'S p :::: 00,1 :::: q :'S (0). Then 2 The HP Norms for Semimartingales 247 Proof. Let H· Z denote (J~ HsdZs)t?o. Recall that we always assume Zo = 0 a.s., and let Z = M +A be a decomposition of Z with Mo = Ao = 0 a.s. Then H· M + H· A is a decomposition of H· Z. Hence IIH· Zlllr ::; jr(H . M, H . A). Next recall that [H· M, H· M] = JH;d[M, M]s, by Theorem 29 of Chap. II. Therefore Jr(H· M,H· A) = 11(100 H;d[M,M]s)1/2 +100 IHslldAslli u ::; IIH~([M,M]~2 +100 IdAsl)llu ::; IIH~IILPII([M,M]~2 +100 IdAsl)IILq = IIHIIQPjq(M, A), where the last inequality above follows from Holder's inequality. The foregoing implies that IIH· ZIIM;r ::; II H llgpjq(M,A) for any such decomposition Z = M + A. Taking infimums over all such de- compositions yields the result. 0 For a process X E JI]) and a stopping time T, recall that X T = X t l[o,T) + XTl[T,oo) , X T - = X t l[o,T) + XT_l[T,oo)' A property holding locally was defined in Chap. I, and a property holding prelocally was defined in Chap. IV. Recall that a property 7r is said to hold locally for a process X if XT n l{Tn>o} has property 7r for each n, where Tn is a sequence of stopping times tending to 00 a.s. If the process X is zero at zero (Le., X o = 0 a.s.) then the property 7r is said to hold prelocally if X Tn - has property 7r for each n. Definition. A process X is locally in S,P (resp. HP) if there exist stopping times (Tn)n?l increasing to 00 a.s. such that XTn l{Tn>o) is in gP (resp. HP) for each n, 1 ::; p ::; 00. IfXo = 0 then X is said to be prelocally in S,P (resp. HP) if X Tn - is in gP (resp. HP) for each n. - While there are many semimartingales which are not locally in HP, all semimartingales are prelocally in HP. The proof of Theorem 4 belowclosely parallels the proof of Theorem 13 of Chap. IV. Theorem 4. Let Z be a semimartingale (Zo = 0). Then Z is prelocally in HP, 1 ::; p ::; 00. 248 V Stochastic Differential Equations Proof.. By the Fundamental Theorem of Local Martingales (Theorem 25 of Chap. III) and the Bichteler-Dellacherie Theorem (Theorem 43 of Chap. III) we know that for given c > 0, Z has a decomposition Z = M +A, Mo = Ao = 0 a.s., such that the jumps of the local martingale M are bounded by c. Define inductively To = 0, Tk+l = inf{t ~ Tk: [M,M]:/2 +l t IdAsl ~ k + I}. The sequence (Tk)k>l are stopping times increasing to 00 a.s. Moreover ZTk- = (MTk) + (ATk- - ~MTkl[Tk,oo)) = N + C is a decomposition of ZTk-. Also, since [M,M]Tk = [M,M]Tk- + (~MTk)2, we conclude joo(N, C) = II[N,N]~2 +100 IdCslllu>o rTk = 11([M,M]Tk- + (~MTk)2)1/2 + Jo IdCs/IILoo :s; II(k2 + c2 )1/2 + (k + c)IILOO < 00. Therefore ZTk- E H oo and hence it is in HP as well, 1 :s; p :s; 00. o Definition. Let Z be a semimartingale in H oo and let a > O. A finite sequence of stopping times 0 = To :s; Tl :s; '" :s; T;;is said to a-slice Z if Z = ZTk- and II (Z - ZT;) T;+l -II Hoo :s; a, 0 :s; i :s; k - 1. If such a sequence of stopping times exists, we say Z is a-sliceable, and we write Z E 5(a). Theorem 5. Let Z be a semimartingale with Zo = 0 a.s. (i) For a > 0, if Z E 5(a) then for every stopping time T, ZT E 5(a) and ZT- E 5(2a). (ii) For every a > 0, there exists an arbitrarily large stopping time T such that ZT- E 5(a). Proof. Since ZT- = MT +(AT- -~MTl[T,oo)),and since IIZTllHoo :s; IIZllHoo always, one concludes IIZT-II11oo :s; 21/Zlllloo, so that (i) follows~ - Next consider (ii). If semimartingale-; Z and Yare a-sliceable, let T[ and TJ be two sequences of stopping times respectively a-slicing Z and Y. By reordering the points T[ and TJ and using (i), we easily conclude that Z + Y is 8a-sliceable. Next let Z = M + A, Mo = Ao = 0 a.s., with the local martingale M having jumps bounded by the constant (3 = a/24. By the preceding observation it suffices to consider M and A separately. 3 Existence and Uniqueness of Solutions 249 For A, let To = 0, T k+1 = inf{t ~ Tk : J~k IdAsl ~ al8 or J~ IdAsl ~ k}. Then ATk- E S(aI8) for each k, and the stopping times (Tk) increase to 00 a.s. For M, let Ro = 0, Rk+l = inf{t ~ Rk : [M,M]t - [M,M]Rk > (32 or [M, M] t ~ k}. Then M Rk - E HOG, each k, and moreover Hence II(M_MRk)Rk+ 1 -IIHoo ~ 11([M,M]Rk+l - [M,M]Rk)1/2 + I~MRk+lIIILoo = 11((~MRk+l)2 + [M, M]Rk+l- - [M, M]Rk)1/2 + I~MRk+lIIILoo ~ 11((32 + (32)1/2 + (311 Lao = (1 + V2)(3. Thus for each k, MRk- E S((l + V2)(3), and since (3 follows. 3 Existence and Uniqueness of Solutions a/24, the result D In presenting theorems on the existence and uniqueness of solutions of stochas- tic differential equations, there are many choices to be made. First, we do not present the most general conditions known to be allowed; in exchange we are able to give simpler proofs. Moreover the conditions we do give are extremely general and are adequate for the vast majority of applications. For more gen- eral results the interested reader can consult Jacod [103, page 451]. Second, we consider only Lipschitz-type hypotheses and thus obtain strong solutions. There is a vast literature on weak solutions (d., e.g., Stroock-Varadhan [220]). However, weak solutions are more natural (and simpler) when the differen- tials are the Wiener process and Lebesgue measure, rather than general semi- martingales. A happy consequence of our approach to stochastic differential equations is that it is just as easy to prove theorems for coefficients that depend not only on the state X t of the solution at time t (the traditional framework), but on the past history of the process X before t as well. We begin by stating a theorem whose main virtue is its simplicity. It is a trivial corollary of Theorem 7 which follows it. Recall that a process H is in L if it has caglad paths and is adapted. Theorem 6. Let Z be a semimartingale with Zo = 0 and let f : JR+ xnx JR ----7 JR be such that (i) for fixed x, (t, w) f---+ f(t, W, x) is in L; and 250 V Stochastic Differential Equations (ii) for each (t, w), If(t, w, x)- f(t, w, y)1 ::::; K(w)lx-yl for some finite random variable K. Let X o be finite and Fo measurable. Then the equation admits a solution. The solution is unique and it is a semimartingale. Of course one could state such a theorem for a finite number of differentials dZi , 1 ::::; j ::::; d, and for a finite system of equations. In the theory of (non-random) ordinary differential equations, coefficients are typically Lipschitz continuous, which ensures the existence and the unique- ness of a solution. In stochastic differential equations we are led to consider more general coefficients that arise, for example, in control theory. There are enough different definitions to cause some confusion, so we present all the def- initions here in ascending order of generality. Note that we add, for technical reasons, the non-customary condition (ii) below to the definition of Lipschitz which follows. Definition. A function f : JR+ x JRn ---+ JR is Lipschitz if there exists a (finite) constant k such that (i) If(t,x) - f(t,y)1 ::::; klx - yl, each t E JR+, and (ii) t I---t f(t,x) is right continuous with left limits, each x E JRn. f is said to be autonomous if f(t,x) = f(x), all t 2: o. Definition. A function f : JR+ x n x JRn ---+ JR is random Lipschitz if f satisfies conditions (i) and (ii) of Theorem 6. Let JI])n denote the space of processes X = (X1, .•. , X n ) where each Xi E JI]) (l::::;i::::;n). Definition. An operator F from JI])n into JI])1 = JI]) is said to be process Lipschitz if for any X, Y in JI])n the following two conditions are satisfied: (i) for any stopping time T, X T- = y T- implies F(Xf- = F(Yf-, and (ii) there exists an adapted process K E L such that Definition. An operator F mapping JI])n to JI])1 = JI]) is functional Lipschitz if for any X, Y in JI])n the following two conditions are satisfied: (i) for any stopping time T, X T- = y T- implies F(X)T- = F(Yf-, and (ii) there exists an increasing (finite) process K = (Kt)t>o such that W(X)t- F(Y)tl ::::; KtllX - YII; a.s., each t 2: O. - 3 Existence and Uniqueness of Solutions 251 Note that if g(t,x) is a Lipschitz function, then f(t,x) = g(t-,x) is ran- dom Lipschitz. A Lipschitz, or a random Lipschitz, function induces a process Lipschitz operator, and if an operator is process Lipschitz, then it is also functional Lipschitz. An autonomous function with a bounded derivative is Lipschitz by the Mean Value Theorem. If a function f has a continuous but not bounded derivative, f will be locally Lipschitz; such functions are defined and con- sidered in Sect. 7 of this chapter. Let A = (Atk,:o be continuous and adapted. Then a linear coeffi- cient such as f(t,w,x) = At(w)x is an example of a process Lipschitz co- efficient. A functional Lipschitz operator F will typically be of the form F(X) = f(t, w; X s, s ::; t), where f is defined on [0, t] x n x D[O, t] for each t 2: 0; here D[O, t] denotes the space of cadlag functions defined on [0, tJ. An- other example is a generalization of the coefficients introduced by Ito and Nisio [101], namely for a random signed measure J.L and a bounded Lipschitz function 9 with constant C(w). In this case, the Lipschitz process for F is given by Kt(w) = C(w)IIJ.L(wM, where 1IJ.L(whll denotes the total mass of the measure J.L(w,du) on [O,t]. Lemmas 1 and 2 which follow are used to prove Theorem 7. We state and prove them in the one dimensional case, their generalizations to n dimensions being simple. Lemma 1. Let 1 ::; p < 00, let J E S-P, let F be functional Lipschitz with F(O) = 0, and suppose SUPt IKt(w)1 ?;k a.s. Let Z be a semimartingale in HOC! such that IIZllMoo ::; 2c~k' Then the equation Xt= Jt + it F(X)s_dZs has a solution in gP. It is unique, and moreover IIXllgP ~ 211 Jllgp· Proof. Define A : gP ----7 gP by A(X)t = Jt + J~ F(X)sdZs. Then by Theo- rems 2 and 3 the operator is 1/2 Lipschitz, and the fixed point theorem gives existence and uniqueness. Indeed IIXllgp ::; IIJllgP + II JF(X)s_dZsllgP ::; IIJllsp + cpllF(X)llspIIZIlHoo = = = 1~ IIJllgp + 2k IIF(X)llgp· 252 V Stochastic Differential Equations Since IIF(X)IIQp = IIF(X) - F(O)IIQP, we have IIXIIQP ~ IIJIIQP + !IIXIIQP, which yields tne estimate. - - - -0 Lemma 2. Let 1 ~ p < 00, let J E S-P, let F be functional Lipschitz with F(O) = 0, and suppose SUPt IKt(w)1 ~k < 00 a.s. Let Z be a semimartingale such that Z E S(2C~k)' Then the equation Xt= Jt +it F(X)s_dZs has a solution ingP. It is unique, and moreover IIXllgP ~ C(k, Z)/IJllgP, where C(k, Z) is a constant depending only on k and Z. Proof Let z = IIZILHoo and j = IIJIIQP. Let °= To,Tl, ... ,Te be the slicing times for Z, and consider the equations, indexed by i = 0,1,2, ... , i=0,1,2, .... (i) Equation (i) has the trivial solution X == °since JO- = Zo- = °for all t, and its gP norm is 0. Assume that equation (i) has a unique solution Xi, and let Xi = IIXiIIQP. Stopping next at Ti instead of Ti-, let yi denote the unique solution of-yi = JTi + JF(yi)s_dz'[i, and set yi = lIyillgp. Since yi = Xi + {~JTi + F(Xi)Ti-~ZTJ1[Ti'oo), we conclude that IlYillgP ~ IIXillgp + 211JIIgP + IIF(Xi)llgPIIZlltboo ~xi+2j+kxiZ = x i (l + kz) + 2j; hence yi ~ 2j + x i(l + kz). (*) We set for U E JI)), DiU = (U - UTifi+l-. Since each solution X of equation (i + 1) satisfies XTi = yi on [0, Ti+d, we can change the unknown by U = X - (yifi+l-, to get the equations U = DiJ+JF(yi +U)s-dDiZs' However since F(yi +0) need not be °we define Gi(·) = F(yi +.) - F(yi), and thus the above equation can be equivalently expressed as We can now apply Lemma 1 to this equation to find that it has a unique solution in gP, and its norm u i is majorized by u i ~ 211 DiJ +JF(yi)s_dDiZsllgP . 1 . ~ 2(2j + epky' 2c p k) ~ 4j + y'. 3 Existence and Uniqueness of Solutions 253 We conclude equation (i + 1) has a unique solution in!iP with norm xi+1 dominated by (using (*)) - xi+ 1 ::::; ui + yi ::::; 4j + 2yi ::::; 8j + 2(1 + kz)xi . Next we iterate from i = °to £ - 1 to conclude that x f < 8{ (2 + 2kz/ - I} .. - 1 + 2kz J Finally, since Z = ZTt -, we have seen that the equation X = J+JF(X)s_dZs has a unique solution in gP, and moreover X = X f + J - JTt-. Therefore IIXIIQP ::::; xl + 2j, and hence C(k, Z) ::::; 2 + 8{ (2i~;k:-1}. D Theorem 7. Given a vector of semimartingales Z = (Zl, ... , Zd), Zo = ° processes Ji E JI]), 1 ::::; i ::::; n, and operators FJ which are functional Lipschitz (1 ::::; i ::::; n, 1 ::::; j ::::; d), the system of equations d t X; = J; + z=1FJ(X)s-dZ~ j=l 0 (1 ::::; i ::::; n) has a solution in JI])n, and it is unique. Moreover if (Ji)iSon is a vector of semimartingales, then so is (Xi)i T, that X(R)T- = X(T)T-, and therefore we can define a process X on n x [0,00) by X = X(T) on [0, T). Thus we have existence. 254 V Stochastic Differential Equations Suppose next Y is another solution. Let S be arbitrarily large such that (X - y)S- is bounded, and let R = min(S, T), which can also be taken arbitrarily large. Then X R- and Y R- are both solutions of and since ZR- E S(2Jsk)' we know that XR- = yR- by the uniqueness established in Lemma 2. Thus X = y, and we have uniqueness. We have assumed that maxi,j SUPt K;,j(w) :::; k < 00 a.s. By proving existence and uniqueness on [0, to], for to fixed, we can reduce the Lips- chitz processes Ki,j to the random constants Ki~j(w), which we replace with K (w) = maxi,j K:~j(w). Thus without loss of generality we can assume we have a Lipschitz constant K(w) < 00 a.s. Then we can choose a constant c such that P(K :::; c) > O. Let On = {K :::; c + n}, each n = 1,2,3, .... Define a new probability Pn by Pn(A) = P(A n On)/P(On), and note that Pn « P. Moreover for n > m we have Pm « Pn. From now on assume n > m in the rest of this proof. Therefore we know that all P semimartingales and all Pn semimartingales are Pm semimartingales, and that on Om a stochastic inte- gral calculated under Pm agrees with the the same one calculated under Pn , by Theorem 14 of Chap. II. Let yn be the unique solution with respect to Pn which we know exists by the foregoing. We conclude yn = ym on Om, a.s. (dPm ). Define 00 ~ = 2: Yt1{lln\lln_d n=l and we have Y = yn a.s. (dPn ) on On, and hence also a.s. (dP) on On, each n. Since 0 = U~=l (On \ On-I) a.s. (dP), we have that on On: yt = Jt +I t F(yn)s_dZs = Jt + it F(Y)s_dZs a.s. (dP), for each n. This completes the proof. D Theorem 7 can be generalized by weakening the Lipschitz assumption on the coefficients. If the coefficients are Lipschitz on compact sets, for example, in general one has unique solutions existing only up to a stopping time T; at this time one has lim SUPt--->T IXt I = 00. Such times are called explosion times, and they can be finite or infinite. Coefficients that are Lipschitz on compact sets are called locally Lipschitz. Simple cases are treated in Sect. 7 of this chapter (d., Theorems 38, 39, and 40), where they arise naturally in the study of flows. 3 Existence and Uniqueness of Solutions 255 We end this section with the remark that we have already met a funda- mental stochastic differential equation in Chap. II, that of the stochastic exponential equation There we obtained a formula for its solution (thus a fortiori establishing the existence of a solution), namely X t = Xoexp{Zt - ~[Z, Z]t} II (1 + ~Zs) exp{-~Zs + ~(~Zs)2}. o 256 V Stochastic Differential Equations II(Xm+l - Xfl-IIs2 :::; ~11(xm - Xfl-lls2 = 2 = :s; 2~ II(X I - X)T1 -11g;2 so that limm--->oo II(Xm+l - X)T1-II.Q2 = O. We next analyze the jump at Tl . Since - Xr,+l = Xr,:!:l + F(Xm)T1_D..ZT1, we have II(Xm+l -xf111g;2:S; II(Xm+l _X)T1-11g;2 +azll(Xm -Xfl-IIg;2, where z = IIZllMoo < 00. Therefore II(Xm+l - X)T111g;2 :s; 2~-1 (1 + az)II(X l - X)T1-11g;2. Next suppose we know that II(Xm+l _ xf£ 11g;2 :s; (m + 1)::~~ + az)£'Y where l' = II(X I - X)Tk -IIg;2. Then (X;nH - Xtf£+l- = (X;n+l _ Xt)T£ + {t (F(Xm)s_ - F(X)s_)dZ~+liT £ where Zf+l = (Z - zT£f£+l-. Therefore by iterating on £, °:s; £ :s; k, II(Xm+l - Xf£+1-11g;2 :s; II(Xm+l - xf£ 11g;2 + aV8o:lI(Xm - Xf£+1-11g;2 :s; (m + 1)~:~(11 + az)£ + aJ8o:ll(Xm _ Xf£+1-11g;2 < (m + 1)£-1(1 + az)£'Y - 2m - l Note that the above expression tends to °as m tends to 00. Therefore x m tends to X prelocally in g2 by a (finite) induction, and hence limm--->oo X m = X in ucp. It remains to remove the assumption that SUPt K t :s; a < 00 a.s. Fix a t < 00; we will show ucp on [0, t]. Since t is arbitrary, this will imply the result. As we did at the end of the proof of Theorem 7, let c > °be such that P(Kt :s; c) > 0. Define f2n = {w : Kt(w) :s; c + n}, and Pn(A) == P(AIf2n). Then Pn « P, and under Pn, limm--->oo X m = X in ucp on [0, t]. For c > 0, choose N such that n 2: N implies P(f2~) < c. Then hence limm--->oo P( (X m - X); > 0) :s; c. Since c > °was arbitrary, the limit must be zero. D 4 Stability of Stochastic Differential Equations 257 4 Stability of Stochastic Differential Equations Since one is never exactly sure of the accuracy of a proposed model, it is important to know how robust the model is. That is, if one perturbs the model a bit, how large are the resulting changes? Stochastic differential equations are stable with respect to perturbations of the coefficients, or of the initial conditions. Perturbations of the differentials, however, are a more delicate matter. One must perturb the differentials in the right way to have stability. Not surprisingly, an HP perturbation is the right kind of perturbation. In this section we will be concerned with equations of the form where I n, J are in J!)), zn, Z are semimartingales, and F n, F are functional Lipschitz, with Lipschitz processes K n , K, respectively. We will assume that the Lipschitz processes K n , K are each uniformly bounded by the same con- stant, and that the semimartingale differentials zn, Z are always zero at 0 (that is, Z[; = 0 a.s., n ~ 1, and Zo = 0 a.s.). For simplicity we state and prove the theorems in this section for one equation (rather than for finite systems), with one semimartingale driving term (rather than a finite number), and for p = 2. The generalizations are obvious, and the proofs are exactly the same except for notation. We say a functional Lipschitz operator F is bounded if for all H E J!)), there exists a non-random constant c < 00 such that F(H)* < c. Theorem 9. Let J, In E J!)); Z, zn be semimartingales; and F, F n be func- tional Lipschitz with constants K, K n , respectively. Assume that (i) J, r are in 5-2 (resp. H 2 ) and limn--->oo r = J in 5-2 (resp. H 2 ); (ii) F n are all bOUnded by the same constant c, and lim~--->oo Fn(X) = F(X) in 5-2 , where X is the solution of (*); and (iii) m~(suPnKn,K):::; a < 00 a.s. (a not random), Z E S(2JsJ, (znk::~l . H 2 d l' zn Z· H 2 2are zn = ,an Imn--->oo = zn = . Then limn--->oo xn = X in 8..2 (resp. in H 2 ), where xn is the solution of (*n) and X is the solution of (*). - Proof. We use the notation H· Zt to denote J~ HsdZs, and H· Z to denote the process (H· Zt)r20. We begin by supposing that J, (r)n21 are in !i2 and In converges to J in §}. Then - 2 S (0:) is defined in Sect. 2 on page 248. 258 V Stochastic Differential Equations x - x n = J - r + (F(X) - Fn(X))_ . Z + (Fn(X) - Fn(xn))_ . Z + Fn(xn)_ . (Z _ zn). Let yn = (F(X) - Fn(X))_ . Z + Fn(xn) . (Z - zn). Then X - X n = J - r + yn + (Fn(x) - Fn(xn))_ . Z. (**) For U E JI)) define en by Then en(U) is functional Lipschitz with constant a and en(O) = O. Take U = X - X n , and (**) becomes By Lemma 2 preceding Theorem 7 we have Since C(a, Z) is independent of nand limn --+ oo IIJ - rll~2 = 0 by hypothesis, it suffices to show limn --+ oo II yn 11~2 = O. But lI yn n§;2 :::; II(F(X) - Fn(X))_ . ZII~2 + IIFn(xn)_ . (Z - zn)II~2 :::; v8IIF(X) - Fn(X)II~21IZIIl!oo (* * *) + v8IIFn(xn)_II~oo liZ - znl11e by Theorem 2 and Emery's inequality (Theorem 3). Since IIZIIHOO < 00 by hypothesis and since - lim IIF(X) - F n(X)lls2 = lim liZ - Z n ll H2 = 0, n---+CX) = n---+CX) - again by hypothesis, we are done. Note that if we knew In, J E H 2 and that In converged to J in H 2 , then We have seen already that limn --+ oo IIX - xn 1I~2 = 0, hence it suffices to show limn--+ oo lIynll1i2 = O. Proceeding as in (* * *fwe obtain the result. 0 Comment. Condition (ii) in Theorem 9 seems very strong; it is the per- turbation of the semimartingale differentials that make it necessary. In- deed, the hypothesis cannot be relaxed in general, as the following example shows. We take n = [0,1], P to be Lebesgue measure on [0,1], and (.rt)t>o equal to .r, the Lebesgue sets on [0,1]. Let ¢(t) = min(t,I), t 2 O. L~t fn(w) 2 0 and set Z['(w) = ¢(t)fn(w), w E [0,1], and Zt(w) = ¢(t)f(w). 4 Stability of Stochastic Differential Equations 259 Let Fn(X) = F(X) == X, and finally let Jt' = Jt = 1, all t 2: o. Thus the equations (*n) and (*) become respectively X~ = 1 +it X~_dZ~, X t = 1 + it Xs_dZs, which are elementary continuous exponential equations and have solutions X~ = exp{Zn = exp{fn(w)¢(t)}, X t = exp{Zd = exp{f(w)¢(t)}. We can choose fn such that limn--->oo E{f~} = 0 but limn--->oo E{f~} =1= 0 for p > 2. Then the zn converge to 0 in H 2 but xn does not converge to X = 1 (since f = 0) in ~P, for any p 2: 1. Indeed, limn--->oo E{f~} =1= 0 for p > 2 implies limn--->oo E{etfn } =1= 1 for any t > o. The next result does not require that the coefficients be bounded, because there is only one, fixed, semimartingale differential. Theorem 10, 11, and 13 all have H 2 as well as ~2 versions as in Theorem 9, but we state and prove only the !j} versions. - Theorem 10. Let J, In E J!)); Z be a semimartingale; F, F n be functional Lipschitz with constants K, K n , respectively; and let X n , X be the unique solutions of equations (*n) and (*), respectively. Assume that (i) I n, J are in ~2 and limn--->oo In = J in ~2; (ii) lim Fn(X) -: F(X) in ~2, where X isthe solution of (*),. and n--+CX) - (iii) max(suPn K n, K) :::; a < 00 a.s. for a non-random constant a, and Z E S(2Jsa)· Then limn--->oo x n = X in ~2 where x n is the solution of (*n) and X is the solution of (*). - Proof. Let x n and X be the solutions of equations (*n) and (*), respectively. Then We let yn = (F(X) - Fn(x))_ . Z, and we define a new functional Lipschitz operator en by Then en(O) = o. If we set U = X - xn, we obtain the equation U = J - r + yn + en(U)_ . Z. 260 V Stochastic Differential Equations Since Z E H=, by Emery's inequality (Theorem 3) we have yn ----> 0 in H 2 , and hence also in g2 (Theorem 2). In particular Ilynll~2 < 00, and therefore by Lemma 2 in Sect. 3 we have - where C(a, Z) is independent of n, and where the right side tends to zero as n ----> 00. Since U = X - X n , we are done. 0 We now wish to localize the results of Theorems 9 and 10 so that they hold for general semimartingales and exogenous processes I n , J. We first need a definition, which is consistent with our previous definitions of properties holding locally and prelocally (defined in Chap. IV, Sect. 2). Definition. Processes M n are said to converge locally (resp. prelocally) in 5.-P (resp. HP) to M if Mn, M are in 5.-P (resp. HP) and if there exists a sequence of stopping times Tk increasing to00 a.s. such that limn--->= II(Mn - M)Tk1{Tk>O} II~p = a (resp. limn--->= II(Mn - Mfk-ll~p = 0) for each k 2 1 (resp. gP repla~ed by HP). - Theorem 11. Let J, In E J!)); Z be a semimartingale (Zo = 0); and F, F n be functional Lipschitz with Lipschitz processes K, K n, respectively. Let X n , X be solutions respectively of X;' = J;' + it Fn(xn)s_dZ" Assume that (i) In converge to J prelocally in 5.-2 ; (ii) Fn(X) converges to F(X) prekJcally ing2 where X is the solution of(*); and (iii) max(suPn K n, K) :::; a < 00 a.s. (a not random). Then limn--->= xn = X prelocally in 5.-2 where X n is the solution of (*n) and X is the solution of (*). - Proof. By stopping at T - for an arbitrarily large stopping time T we can assume without loss of generality that Z E S( 2Jsa) by Theorem 5, and that In converges to J in g2 and F(Xn) converges to F(X) in g2, by hypothesis. Next we need only to apply Theorem 10. 0 We can recast Theorem 11 in terms of convergence in ucp (uniform conver- gence on compacts, in probability), which we introduced in Sect. 4 of Chap. II in order to develop the stochastic integral. 4 Stability of Stochastic Differential Equations 261 Corollary. Let I n , J E J!)); Z be a semimartingale (Zo = 0); and P, p n be functional Lipschitz with Lipschitz processes K, K n, respectively. Let X, xn be as in Theorem 11. Assume that (i) In converges to J in ucp, (ii) pn(x) converges to P(X) in ucp, and (iii) max(suPn K n , K) :::; a < 00 a.S. (a not random). Then lim X n = X in ucp. n--+oo Proof Recall that convergence in ucp is metrizable; let d denote a distance compatible with it. If xn does not converge to 0 in ucp, we can find a subse- quence n' such that infn, d(xn', 0) > O. Therefore no sub-subsequence (xn ll ) can converge to 0 in ucp, and hence X n" cannot converge to 0 prelocally in !i2 as well. Therefore to establish the result we need to show only that for ;:-ny subsequence n', there exists a further subsequence nil such that X n " con- verges prelocally to 0 in !i2 . This is the content of Theorem 12 which follows, so the proof is complete.- 0 Theorem 12. Let Hn, H E J!)). Por Hn to converge to H in ucp it is necessary and sufficient that there exist a subsequence n' such that limn' --+00 Hn' = H, prelocally in g2. Proof We first show the necessity. Without loss of generality, we assume that H = O. We construct by iteration a decreasing sequence of subsets (Nk) of N = {I, 2, 3, ... }, such that lim sup IH~I = 0 a.s. ~EN~O~S~k By Cantor's diagonalization procedure we can find an infinite subset N' of N such that lim sup IH;'I = 0 a.s., r:.~w O~s9 each integer k > O. By replacing N with N' we can assume without loss of generality that Hn tends to 0 uniformly on compacts, almost surely. We next define Tn = inf{t ~ 0: IHI'I ~ I}, Sn = inf Tm .m~n Then Tn and Sn are stopping times and the Sn increase to 00 a.S. Indeed, for each k there exists N(w) such that for n ~ N(w), sUPO 262 V Stochastic Differential Equations Then (L ffi )*:::; sup IH~I, O~s~n which tends to 0 a.s. Moreover, when m 2 n, (Lffi )*:::; sup IH~I:::; sup IH~I:::; 1. O~S 0, E: > 0, and a subsequence n' such that P{(Hn ' - H);o > 8} > E: for each n'. Then II(Hn' - H);oll£2 2 8Vi for all n'. Let T k tend to 00 a.s. such that (Hn' - H)T k - tends to 0 in rj}, each k. Then there exists K > 0 such that P(Tk < to) < Â¥ for all k > K. Hence We conclude 8Vi :::; Â¥, a contradiction. Whence Hn converges to H in u~. 0 Recall that we have stated and proven our theorems for the simplest case of one equation (rather than finite systems) and one semimartingale driving term (rather than a finite number). The extensions to systems and several driving terms is simple and essentially only an exercise in notation. We leave this to the reader. An interesting consequence of the preceding results is that prelocal fiP and prelocal HP convergence are not topological in the usual sense. If they were, then onewould have that a sequence converged to zero if and only if every subsequence had a sub-subsequence that converged to zero. To see that this is not the case for fi2 for instance, consider the example given in the comment following Theorem 9. In this situation, the solutions X n converge to X in u~. By Theorem 12, this implies the existence of a subsequence n' such that limn,-->oo X n' = X, prelocally in fi2. However we saw in the comment that x n does not converge to X in fi2~ It is still a priori possible that x n converges to X prelocally in fi2, how;'er. In the framework of the example a stopping time is simply a non-negative random variable. Thus our counter- example is complete with the following real analysis result (see Protter [199, page 344] for a proof). There exist non-negative functions in on [0,1] such that limn-->oo J01in(x)2dx = 0 and limsuPn-->oo JA (fn (x))Pdx = +00 for all p > 2 and all Lebesgue sets A with strictly positive Lebesgue measure. In conclusion, this counterexample gives a sequence of semimartingales xn such that every subsequence has a sub-subsequence converging prelocally in fiP , but the sequence itself does not converge prelocally in gP, (1:::; p < 00). - 4 Stability of Stochastic Differential Equations 263 Finally, we observe that such non-topological convergence is not as un- usual as one might think at first. Indeed, let X n be random variables which converge to zero in probability but not a.S. Then every subsequence has a sub- subsequence which converges to zero a.s., and thus almost sure convergence is also not topological in the usual sense. As an example of Theorem 10, let us consider the equations Xf = Jf + it Fn(xn)sdWs + it cn(xn)sds, Xt= Jt+ it F(X)sdWs + it C(X)sds where W is a standard Wiener process. If (In - J); converges to 0 in L 2 , each t> 0, and if F n, cn, F, C are all functional Lipschitz with constant K < 00 and are such that (Fn(X) - F(X))t and (Cn(X) - C(X))t converge to 0 in L 2 , each t > 0, then (xn - X); converges to 0 in L 2 as well, each t > O. Note that we require only that Fn(X) and Cn(X) converge respectively to F(X) and C(X) for the one X that is the solution, and not for all processes in ]])). One can weaken the hypothesis of Theorem 9 and still let the differentials vary, provided the coefficients stay bounded, as the next theorem shows. Theorem 13. Let I n, J E]])); Z, zn be semimartingales (Zr; = Zo = 0 a.s.); and F, F n be functional Lipschitz with Lipschitz processes K, K n, respectively. Let X n, X be solutions of (*n) and (*), respectively. Assume that (i) r converges to J prelocally in §:.2; (ii) Fn(X) converges to F(X) prelcJcally in §:.2, and the coeffiCients F n , F are all bounded by c < 00; - (iii) zn converges to Z prelocally in H 2 ; and (iv) max(suPn K n, K) :::; a < 00 a.s. (a not random). Then lim X n = X prelocally in §:.2. n-+CX) - Proof By stopping at T - for an arbitrarily large stopping time T we can assume without loss that Z E S(2Js-a) by Theorem 5, and that In converges to J in §:.2, Fn(X) converges in §:.2 to F(X), and zn converges to Z in H 2 , all by hypothesis. We then invokeTheorem 9, and the proof is complete. - 0 The assumptions of prelocal convergence are a bit awkward. This type of convergence, however, leads to a topology on the space of semimartingales which is the natural topology for convergence of semimartingale differentials, just as ucp is the natural topology for processes related to stochastic integra- tion. This is exhibited in Theorem 15. Before defining a topology on the space of semimartingales, let us recall that we can define a "distance" on]])) by setting, for Y, Z E ]])) 264 V Stochastic Differential Equations r(Y) = L 2-n E{l/\ sup lÂ¥tll, n>O O::;t::;n and d(Y, Z) = r(Y - Z). This distance is compatible with uniform convergence on compacts in probability, and it was previously defined in Sect. 4 of Chap. II. Using stochastic integration we can define, for a semimartingale X, f(X) = sup r(H. X) IHI::;l where the supremum is taken over all predictable processes bounded by one. The semimartingale topology is then defined by the distance d(X, Y) = f(X - Y). The semimartingale topology can be shown to make the space of semi- martingales a topological vector space which is complete. Furthermore, the fol- lowing theorem relates the semimartingale topology to convergence in HP. For its proof and a general treatment of the semimartingle topology, see Emery [66] or Protter [201]. Theorem 14. Let 1 :::; p < 00, let x n be a sequence of semimartingales, and let X be a semimartingale. (i) If xn converges to X in the semimartingale topology, then there exists a subsequence which converges prelocally in HP. (ii) If x n converges to X prelocally in HP, then it converges to X in the semimartingale topology. - In Chap. IV we established the equivalence of the norms IIXIIlf2 and sUPIHI::;l IIH .XII~2 in the corollary to Theorem 24 in Sect. 2 ofthat chapter. Given this result,-Theorem 14 can be seen as a uniform version of Theorem 12. We are now able once again to recast a result in terms of ucp convergence. Theorem 13 has the following corollary. Corollary. Let I n, J E J!)); zn, Z be semimartingales (Z[; = Zo = 0); and F n , F be functional Lipschitz with Lipschitz processes K, K n , respectively. Let xn, X be solutions of (*n) and (*), respectively. Assume that (i) In converges to J in ucp, (ii) Fn(X) converges to F(X) in ucp where X is the solution of (*), and moreover all the coefficients F n are bounded by a random c < 00, (iii) zn converges to Z in the semimartingale topology, and (iv) max(suPn K n , K) :::; a < 00 a.s. Then lim X n = X in ucp. n->(x> Proof Since zn converges to Z in the semimartingale topology, by Theo- rem 14 there exists a subsequence n' such that zn' converges to Z prelocally in H 2 . Then by passing to further subsequences if necessary, by Theorem 12 wemay assume without loss that In converges to J and Fn(X) converges 4 Stability of Stochastic Differential Equations 265 to F(X) both prelocally in ~2, where X is the solution of (*). Therefore, by Theorem 13, X n converges to X prelocally in ~2 for this subsequence. We have shown that for the sequence (xn) there is always a subsequence that converges prelocally in g2. We conclude by Theorem 12 that xn converges to Xinu~. 0 The next theorem extends Theorem 9 and the preceding corollary by re- laxing the hypotheses on convergence and especially the hypothesis that all the coefficients be bounded. Theorem 15. Let I n , J E J!)); zn, Z be semimartingales (Zr; = Zo = 0); and F n, F be functional Lipschitz with Lipschitz process K, the same for all n. Let xn, X be solutions respectively of Xl' = Jr + l t Fn(xn)s_dZ~, Assume that (i) In converges to J in ucp; (ii) Fn(X) converges to F(X) in ucp, where X is the solution of (*); and (iii) zn converges to Z in the semimartingale topology. Then x n converges to X in ucp. Proof First we assume that SUPt Kt(w) :::; a < 00. We remove this hypothesis at the end of the proof. By Theorem 12, it suffices to show that there exists , 2 a subsequence n' such that xn converges to X prelocally in ~ . Then by Theorem 12 we can assume with loss of generality, by passing to asubsequence if necessary, that In converges to J and Fn(X) converges to F(X) both prelocally in ~2. Moreover by Theorem 14 we can assume without loss, again by passing to-a subsequence if necessary, that zn converges to Z prelocally in H 2 , and that Z E S(4Js-). Thus all the hypotheses of Theorem 13 are satisfied except one. We do not assume that the coefficients F n are bounded. However by pre-stopping we can assume without loss that IF(X)I is uniformly bounded by a constant c < 00. Let us introduce truncation operators T X defined (for x ~ 0) by TX(Y) = min(x,sup(-x, Y)). Then T X is functional Lipschitz with Lipschitz constant 1, for each x > o. Consider the equations 266 V Stochastic Differential Equations Then, by Theorem 13, yn converges to X prelocally in !i2 . By passing to yet another subsequence, if necessary, we may assume that pn(x) tends to F(X) and yn tends to X uniformly on compacts almost surely. Next we define The stopping times Sk increase a.S. to 00. By stopping at Sk_, we have for n ;::: k that (Yn - X)* and (Fn(x) - F(X))* are a.S. bounded by 1. (Note that stopping at Sk_ changes Z to being in S(2)sa) instead of S(4)sa) by Theorem 5.) Observe that IFn(Yn)1 :::; IFn(Yn) - Fn(X)/ + IFn(x) - F(X)I + IF(X)I :::; a(yn - X)* + (Fn(x) - F(X))* + F(X)* :::; a + 1 + c, whence (Ta+c+1Fn)(yn) = Fn(yn). We conclude that, for an arbitrarily large stopping time R, with In and zn stopped at R, yn is a solution of which is equation (*n). By the uniqueness of solutions we deduce yn = xn on [0, R). Since yn converges to X prelocally in !i2 , we thus conclude xn converges to X prelocally in !i2 . - It remains only to removethe hypothesis that SUPt Kt(w) :::; a < 00. Since we are dealing with local convergence, it suffices to consider sUPs a be such that P(Kt :::; a) > 0, and define nm = {w : Kt (w) :::; a + m}. Then nm increase to n a.s. and as in the proof of Theorem 7 we define Pm by Pm(A) = p(Alnm), and define:F[' = Ftb", , the trace of Ft on nm . Then Pm ~ P, so that by Lemma 2 preceding Theorem 25 in Chap. IV, if zn converges to Z prelocally in H 2 (P), then zn converges to Z prelocally in H 2 (Pm ) as well. Therefore by the first part of this proof, xn converges to X in ucp under Pm, each m ;::: 1. Choose c > a and m so large that p(n~J < c. Then lim P((Xn - X); > 8):::; lim Pm((Xn - X); > 8) + p(n~) n---+oo n--+oo :::; lim Pm((Xn - X); > 8) + c n-HX> and since c > awas arbitrary, we conclude that xn converges to X in ucp on [0, t]. Finally since t was arbitrary, we conclude xn converges to X in ucp. 0 4 Stability of Stochastic Differential Equations 267 Another important topic is how to approximate solutions by difference solutions. Our preceding convergence results yield two consequences (Theo- rem 16 and its corollary). The next lemma is a type of Dominated Convergence Theorem for stochas- tic integrals and it is used in the proof of Theorem 16. Lemma (Dominated Convergence Theorem). Let p, q, r be given such that ~ + ~ = ~, where 1 < r < 00. Let Z be a semimartingale in Hq, and let Hn E $..P such that IHnl ::;; Y E $..P, all n 2:: 1. Suppose limn--->oo Hl'_(w) = 0, all (t,W). Then - nl~~ II JH:_dZsIIMT = O. Proof. Since Z E Hq, there exists a decomposition of Z, Z = N + A, such that jq(N,A) = II[N, N]~2 +100 IdAsIII Lq < 00. Let en be the random variable given by en = (100 (H:_)2d[N, N]s)1/2 +100 IH:-1ldAsl. The hypothesis that IHnl ::;; Y implies en ::;; Y* ([N, N]~2 +100 IdAsI) a.s. However IIY*([N,N]~2 +100 IdAsl)llLT ::;; IIY*IILPII[N,N]~2 +100 IdAsIIILq = 1IYIIgpjq(N,A) < 00. Thus en is dominated by a random variable in L T and hence by the Dominated Convergence Theorem en tends to 0 in LT. 0 We let an denote a sequence of random partitions tending to the identity.3 Recall that for a process Y and a random partition a = {O = To ::;; T1 ::;; ... ::;; nn}, we define y a == Yo 1{o} + 2: YT k 1(Tk,Tk+l]' k Note that if Y is adapted, cadlag (i.e, Y E ID», then (~a)s;~o is left continuous with right limits (and adapted, of course). It is convenient to have a version of ya E ID>, occasionally, so we define y a + = LYTk 1[Tk ,Tk+l)' k 3 Random partitions tending to the identity are defined in Chap. II, preceding Theorem 21. 268 V Stochastic Differential Equations Theorem 16. Let J E §..2, let F be process Lipschitz with Lipschitz process K :::; a < 00 a.s. and F(O) E g2. Let Z be a semimartingale in S( 2Js), and let X(a) be the solution of X t = Jt + it F(xa+)~dZs, (M) for a random partition a. If an is sequence of random partitions tending to the identity, then X(an ) tends to X in g2, where X is the solution of (*) of Theorem 15. Proof. For the random partition an (n fixed), define an operator en on]]} by en(H) = F(Han+)an+. Note that en(H) E ]]} for each HE]]} and that en(H)_ = F(Han+)an. Then en is functional Lipschitz with constant K and sends §..2 into itself, as the reader can easily verify. Since F(O) E §..2, so also are en"(O) E §..2, n 2': 1, and an argument analogous to the proof ofTheorem 10 (though a bit simpler) shows that it suffices to show that J~ en(X)s_dZs converges to J~ F(X)s_dZs in §..2, where X is the solution of X t = Jt + it F(X)s_dZs. Towards this end, fix (t, w) with t > 0, and choose E > O. Then there exists o> 0 such that IXu(w) - Xt-(w)1 < E for all U E [t - 20, t). If mesh(a) < 0, then also IX;;+(w) - Xu(w)1 < 2E for all u E [t - 0, t). This then implies that IF(xa+)(w) - F(X)(w)1 < 2aE. Therefore (F(xan+) - F(X))fn(w) tends to o as mesh(an ) tends to O. Since, on (0,00), lim F(x)an = F(X)_, mesh(an)--->O we conclude that lim F(Xan+)an = F(X)_, mesh(an)--->O where convergence is pointwise in (t, w). Thus lim en(X)_ = F(X)_. mesh(an)--->O However we also know that len(X)_1 = IF(Xan+)anl :::; F(Xan+)* :::; F(O)* + K(Xan+)* :::; F(O)* + aX* which is in §..2 and is independent of an' Therefore using the preceding lemma, the Domin~ed Convergence Theorem for stochastic integrals, we obtain the convergence in g2 of J~ en(X)s_dZs to J~ F(X)s_dZs, and the proof is com- plete. 0 4 Stability of Stochastic Differential Equations 269 Remark. In Theorem 16 and its corollary which follows, we have assumed that F is process Lipschitz, and not functional Lipschitz. Indeed, Theorem 16 is not true in general for functional Lipschitz coefficients. Let Jt = l{t2: 1} ' Zt = t 1\ 2, and F(Y) = Y11{t2:1}' Then X, the solution of (*) of Theorem 15 is given by X t = (t 1\ 2) 1{t2: I}, but if a is any random partition such that Tk f 1 a.s., then (X(a)"+)t = 0 for t ::;; 1, and therefore F(X(a)a+) = 0, and X(a)t = Jt = l{t2:1}' (Here X(a) denotes the solution to equation (*a) of Theorem 16.) Corollary. Let J E ID>; F be process Lipschitz; Z be a semimartingale; and let an be a sequence of random partitions tending to the identity. Then lim X(an) = X in ucp n--->oo where X (an) is the solution of (*a) and X is the solution of (*), as in Theo- rem 16. Proof First assume K ::;; a < 00, a.s. Fix t > 0 and E > O. By Theorem 5 we can find a stopping time T such that ZT- E S( sJsa)' and P(T < t) < E. Thus without loss of generality we can assume that Z E S(sJsJ. By letting Sk = inf{t 2: 0 : IJt I > k}, we have that Sk is a stopping time, and limk--->oo Sk = 00 a.s. By now stopping at Sk - we have that J is bounded, hence also in fi2, and Z E S(4Jsa)' An analogous argument gives us that F(O) can be assu~d bounded (and hence in g2) as well; hence Z E S(2}8)' We now can apply Theorem 16 to obtain the result. To remove the assumption that K ::;; a < 00 a.s., we need only apply an argument like the one used at the end of the proofs of Theorems 7, 8 and 15. 0 Theorem 16 and its corollary give us a way to approximate the solution of a general stochastic differential equation with finite differences. Indeed, let X be the solution of where Z is a semimartingale and F is process Lipschitz. For each random partition an = {O = To ::;; Tf ::;; ... ::;; TJ: n }, we see that the random variables X(an)TJ: verify the relations (writing a for an, X for X(an), Tk for TJ:) X To = Jo, X Tk+1 = X Tk + JTk+1 - JTk + F(Xa+)Tk(ZTk+l - ZTk)' Then the solution of the finite difference equation above converges to the solution of (*), under the appropriate hypotheses. As an example we give a means to approximate the stochastic exponential. 270 V Stochastic Differential Equations Theorem 17. Let Z be a semimartingale and let X = £(Z), the stochastic exponential of Z. That is, X is the solution of Let an be a sequence of random partitions tending to the identity. Let kn-l x n = IT (1 + (ZTi+l - zTt)). i=l Then lim x n = X in ucp. n--->oo Proof. Let yn be the solution of yt = 1 + It ysandZs, equation (*a) of Theorem 16. By the corollary of Theorem 16 we know that yn converges to X = £ (Z) in ucp. Thus it suffices to show yn = xn. Let an = {O = To :s: Tr :s: ... :s: TrJ· On (Tin, TItl] we have ytn = yr,n + yr,n(ZTi+l _ zTt) = yr,n(l + (ZTi+l _ ZTin)). Inducting on i down to 0 we have ytn = IT (1 + (ZT7+ 1 - Z T7)), j$.i for Tr < t :s: TItl' Since z T7+1 - ZTjn = a for all j > i when Tr < t :s: TItl' we have that yn = xn, and the theorem is proved. 0 5 Fisk-Stratonovich Integrals and Differential Equations In this section we extend the notion of the Fisk-Stratonovich integral given in Chap. II, Sect. 7, and we develop a theory of stochastic differential equa- tions with Fisk-Stratonovich differentials. We begin with some results on the quadratic variation of stochastic processes. Definition. Let H, J be adapted, cadlag processes. The quadratic covari- ation process of H, J denoted [H, J] = ([H, J]tk::o, if it exists, is defined to be the adapted, cadlag process of finite variation on compacts, such that for any sequence an of random partitions tending to the identity, 5 Fisk-Stratonovich Integrals and Differential Equations 271 lim San(H,J) = lim HoJo + ~(HTI'+l _HTin)(JTI'+l _JTt)n~oo n--+oo L..J i = [H,J] with convergence in ucp, where an is the sequence a= To :s: T{' :s: ... :s: TJ: . A process H in ill> is said to have finite quadratic variation if [H, H]t exists and is finite a.s., each t 2: O. If H, J, and H +J in ill> have finite quadratic variation, then the polarization identity holds: 1[H, J] = 2([H + J, H + J] - [H, H] - [J, J]). For X a semimartingale, in Chap. II we defined the quadratic variation of X using the stochastic integral. However Theorem 22 of Chap. II shows every semimartingale is of finite quadratic variation and that the two definitions are consistent. Notation. For H of finite quadratic variation we let [H, H]C denote the con- tinuous part of the (non-decreasing) paths of [H, H]. Thus, [H,H]t = [H,H]~+ L ~[H,H]s, O~s~t where ~[H, H]t = [H, H]t - [H, H]t-, the jump at t. The next definition extends the definition of the Fisk-Stratonovich integral given in Chap. II, Sect. 7. Definition. Let H E ill>, X be a semimartingale, and assume [H, X] exists. The Fisk-Stratonovich integral of H with respect to X, denoted J; H s - 0 dXs, is defined to be rt t 1io Hs- odXs == ioHs_dXs + 2[H,X]~. To consider properly general Fisk-Stratonovich differential equations, we need a generalization of Ito's formulas (Theorems 32 and 33 of Chap. II). Since Ito's formula is proved in detail there, we only sketch the proof of this generalization. Theorem 18 (Generalized Ito's Formula). Let X = (Xl, ... ,xn) be an n-tuple of semimartingales, and let f : lR+ x n x lRn -t lR be such that (i) there exists an adapted FV process A and a function g such that f(t,w,x) = I t g(s,w,x)dAs, (s,w) t-+ g(s,w,x) is an adapted, jointly measurable process for each x, and J; SUPxEK Ig(s,w,x)lldAsl < 00 a.s. for compact sets K. 272 V Stochastic Differential Equations (ii) the function g of (i) is C2 in x uniformly in s on compacts. That is, n sup{lg(s,w,y) - Lgxi(S,W,X))(Yi - Xi) sSct i=l - L gX;Xj(s,W,X)(Yi -Xi)(Yj -Xj)l} lSci,jScn a.s., where rt : n x 1R+ -t 1R+ is an increasing function with limulO rt(u) = a a.s., provided X ranges through a compact set (rt depends on the compact set chosen). (iii) the partial derivatives fxi' fx;xj' 1 :::; i, j :::; n all exist and are continu- ous, and moreover fx,(t,w,X) = it gx;(s,w,x)dAs , fX;Xj(t,w,x) = it gX;Xj(s,w,x)dAs. Then n -g(s,w,Xs)L\As - Lfx;(s-,w,Xs-)L\X~}. i=l Proof. We sketch the proof for n = 1. We have, letting a = to :::; h :::; ... :::; t m = t be a partition of [0, t], and assuming temporarily IXI :::; k for all s :::; t, k a constant, f(t, w, Xt) - f(O, w, Xo) m-l m-l = L f(tkH,W,Xtk+1 )- f(tk,w,Xtk+1 ) +L f(tk,w,Xtk+1 )- f(tk,w,XtJ k=O k=O m-l1t k +1 m-l= L g(u,w,Xtk+JdAu + L f(tk,w,Xtk+1 ) - f(tk,w,Xtk )k=O tk k=O Consider first the term EZ'~ol fttkk+1 g(u,W,Xtk+1 )dAu . The integrand is not adapted, however one can interpret this integral as a path-by-path Stieltjes 5 Fisk-Stratonovich Integrals and Differential Equations 273 integral since A is an FV process. Expanding the integrand for fixed (u,w) by the Mean Value Theorem yields g(u,W,Xtk+1 ) = g(u,w,Xu) +gx(u,w,XU )(X tk+l -Xu) where Xu is in between Xu and X tk+1 • Therefore and since A is of finite variation and X is right continuous, the second sum tends to zero as the mesh of the partitions tends to zero. Therefore letting 7rn denote a sequence of partitions of [0, t] with limn-wo mesh(7rn) = 0, a.s. Next consider the second term on the right side of (*), namely m-l 2: f(tk,w,Xtk+J - f(tk,w,Xtk )· k=O Here we proceed analogously to the proof of Theorem 32 of Chap. II. Given e > 0, t > 0, let A(e, t) be a set of jumps of X that has a.s. a finite number of times s, and let B = B(e,t) be such that LSEB(L\Xs )2::::; e2 , where AuB exhaust the jumps of X on (0, t]. Then m-l 2:f(tk,W,Xtk+ 1 ) - f(tk,w,Xtk ) k=O = Lf(tk,w,Xtk+1 ) - f(tk,w,XtJ + Lf(tk,w,Xtk+1 ) - f(tk,w,Xtk ) k,A k,B By Taylor's formula, and letting L\kX denote X tk+1 - X tk , 274 V Stochastic Differential Equations L!(tk,W,Xtk+1 ) - !(tk,W,Xtk ) k,B = L!x(tk,W,Xtk)LJ.kX + ~ L!xx(tk,W,Xtk )(LJ.kX )2 k k '" 1 2 - ~!x(tk,W,Xtk)LJ.kX + "2!xx(tk,w,Xtk)(LJ.kX) k,A + LR(tk,W,Xtk,Xtk+l)' k,B (**) By Theorems 21 and 30 of Chap. II, the first sums on the right above con- verge in ucp to J~ !x(s-, w, X s- )dXs and ~ J~ !xx(s-, w, X s- )d[X, X]s, re- spectively. The third sum converges a.s. to - L {fx(s-,w, Xs-)LJ.Xs + ~!xx(s-,w,Xs_)(LJ.Xs)2}. sEA By condition (ii) on the function g, we have limsup L R(tk,W,Xtk,Xtk+l):::; rt(w,E+)[X,X]t n tkE1l"n,B where rt(w,E+) = lim sup rt(w,o). O.J.c: Next we let E tend to 0; then rt(w,E+)[X,X]t tends to 0 a.s., and finally combining the two series indexed by A we see that L {!(s-,w,Xs) - !(s-,w,Xs_) - !x(s-,w,Xs-)LJ.Xs sEA(c:,t) 1 2 - "2!xx(s-,w,Xs-)(LJ.Xs) } tends to the series L {f(s-,w, X s) - !(s-,w,Xs_) - !x(s-,w,Xs-)b.Xs O 5 Fisk-Stratonovich Integrals and Differential Equations 275 Note that a consequence of Theorem 18 is that for f satisfying the hy- potheses, we have f(t,·, Xt) is a semimartingale when X = (Xl, ... , xn) is an n-tuple of semimartingales. Also, an important special case is when the process As == s; in this case the hypotheses partly reduce to assuming f is absolutely continuous in t. The next theorem allows an improvement of the Fisk-Stratonovich change of variables formula given in Chap. II (Theorem 34). Theorem 19. Let X = (Xl, ... , Xd) be a vector of semimartingales and let f : ~+ x n x ~d -t ~ be such that (i) there exists an adapted FV process A and a function g such that f(t,w,x) = I t g(s,w,x)dAs where (s, w) I--> g(s, w, x) is an adapted and jointly measurable process for each x. (ii) for each compact set K, J~ SUPxEK Ig(s,w,x)lldAsl < 00 a.s. (iii) fXi exists and is continuous in x and fXi(t,W,X) = J~ gXi(s,w,x)dAs. Let yt = f(t, W, Xl, ... , X d ). Then Y is an adapted process of finite quadratic variation, and moreover Proof. First assume that for fixed (t, w) the function f and all its first partials are bounded functions of x. Then by optional stopping at times T - we can assume without loss of generality that f and all its first partials (in x) are bounded functions. Let fk be a sequence of functions on ~+ x n x ~d which are C2 on ~d such that fk and its first partials converge uniformly on ~d respec- tively to f and its corresponding first partials. Moreover we can take all the fk bounded and Lipschitz continuous with Lipschitz constant c, independent of k. For simplicity take d = 1. Let an be a sequence of random partitions tending to the identity. We write, for a process Z, where an = {o = To ~ T{' ~ ... ~ 1k'n}· Since fk and their first partials converge to f and its first partials uni- formly, we have !k - h is Lipschitz with constant Eke, which tends to 0 as k, £ tend to 00. Therefore, letting T denote an for a given n and writing fk(X) for fk(t,w,Xt ), we have 276 V Stochastic Differential Equations IST(!k(X)) - ST(!£(X))I ~ ST((!k - !£)(X)) + 2{ST(!k(X))ST((!k - !£)(X))}1/2 (*) ~ (E~" + 2CCk£)SAX) by the Lipschitz properties. Since !k(X) and f,,(X) are semimartingales, we know (by Theorem 22 of Chap. II) that San (!k(X)) and San (!£(X)) converge in ucp respectively to [Jk(X), fk(X)] and [!l'(X) , !£(X)]. By restricting our attention to an interval (s, t], we then have from (*) that l{[fk(X), !k(X)]t - [fk(X), fk(X)]s} - {[f,,(X), !£(X)]t - [!£(X), !l'(X)]s}1 ~ (E~" + 2CCk,,){[X,X]t - [X,X]s}. Since this is true for all 0 < s < t < 00, we deduce that Therefore d[!k(X), fk(X)] is a Cauchy sequence of random measures, con- verging in total variation norm on (0,00), a.s. In addition, by Ito's formula (Theorem 18) and Theorem 29 of Chap. II we have that from Theorem 23 of Chap. II we also know that (A!k(X))2 = A[!k(X), !k(X)]. Therefore [fk(X), fk(X)]t = it 7; (s-,w,Xs_)2d[X,X]~ + 2: (Afk(Xs))2, o o:Ss:st and since we have a.s. convergence in total variation norm, we can pass to the limit to obtain Vi = it f'(Xs_)2d[X,X]~+ 2: (Af(Xs))2. o o:Ss:st Our identification of the limit removes the dependence on the representation of Y by f, because ISan (f(X))t - Vii ~ ISan (f(X))t - San (!k(X))tl + IVi - [!k(X),fk(X)]tl + ISan (!k(X))t - [!k(X), fk(X)]tl, and by taking k large enough the first two terms on the right side can be taken small in probability independently of n; one then takes n large enough in the 5 Fisk-Stratonovich Integrals and Differential Equations 277 third term to make it small in probability, and we have limn ---.oo San (J(X))t = lit, in probability. Therefore lit = [Y, Y]t and this completes the proof for f and its first partials bounded functions of x, and d = 1. The proof for general d is exactly analogous. For general f satisfying the hypotheses, let gm : lR.d ~ lR.d be Coo such that gm(x) = X if Ixl :::; m, and gm has compact support. Defining ~m = f(t, w, gm(Xt )), let Tm(w) = inf{t ~ 0 : l't(w) :f. ~m(w)}. Then Tm increase to 00 a.s. Also, since the quadratic variation is a path property, [ym, ymIt (w) = [Y, Ylt (w), for all t < Tm (w). Therefore Y is of finite quadratic variation, and since by the preceding (for d = 1) [ym,Ymlt =it ~~ (s-,w,gm(Xs_))g~(Xs_)d[X,X]~ + L f1f(s,w,gm(Xs))2 o O 278 V Stochastic Differential Equations Theorem 21. Let X = (xl, ... ,xn) be an n-tuple of semimartingales, and let f : JR.n --> JR. have second order continuous partial derivatives. Then f(X) is a semimartingale and the following formula holds: As a corollary of Theorem 21, we have the Stratonovich integration by parts formula. Corollary 1 (Stratonovich Integration by Parts Theorem). Let X and Y be semimartingales. Then Proof. Let f(x, y) = xy and apply Theorem 21. o Corollary 2. Let X and Y be semimartingales, with at least one of X or Y continuous. Then XtYl - XoYo = t X s- od~ + rt Ys- odXs.Jo+ Jo+ Recall that since X o- = 0 by convention for a dl,dlag process X, we did not really need to write f;+; the formula also holds for f;. For stochastic differential equations with Fisk-Stratonovich differentials we are limited as to the coefficients we can consider, because the integrands must be of finite quadratic variation. Nevertheless we can still obtain reasonably general results. We first describe the coefficients. Definition. A function f : JR.+ x nx JR.d --> JR. is said to be Fisk-Stratonovich acceptable if (i) there exists an adapted FV process A and a function g such that f(t,w,x) = it g(s,w,x)dAs where (s, w) I---' g( S, w, x) is an adapted, jointly measurable process for each x; (ii) for each compact set K, f; sUPxEK Ig(s,w, x)lldAsl < 00 a.s.; (iii) fXi is C1 and 5 Fisk-Stratonovich Integrals and Differential Equations 279 (iv) for each fixed (t, w), the functions x f---4 f(t, w, x) and x f---4 (j!,. J)(t, w, x) are all Lipschitz continuous with Lipschitz constant K(w), K < 00 a.s. (1 :::; i :::; d).4 We will often write "F-S acceptable" in place of "Fisk-Stratonovich ac- ceptable." Theorem 22. Given a vector of semimartingales Z = (Zl, ... , Zk), semi- martingales Ji (1 :::; i :::; d), and F-S acceptable junctions fj (1 :::; i :::; d,l :::; j :::; k), then the system of equations k t X; = Ji + L1fj(s-,w,X s-) odZ~ j=l 0 has a unique semimartingale solution. Moreover the solution X of (*) is also the (unique) solution of Proof We note that equation (**) has a unique solution as a trivial conse- quence of Theorem 7. Since X is ad-dimensional semimartingale, we know that f;(s-,., X s-) is in the domain of definition of the F-S integral by The- orem 19. Further, as a consequence of Theorem 19, we have that5 and the equivalence of (*) and (**) follows. Therefore the existence of a unique semimartingale solution of (**) is equivalent to the existence of a unique semimartingale solution of (*), and we are done. Note that if Jm is of finite variation, the terms involving [Jm, Zj]C disappear. 0 4 By (~. f)(t,w,x) we mean the usual product offunctions ~(t,w, x)· f(t,w,x). 5 Since we are taking the continuous parts of the quadratic variations, we need not write f(8-, w, X s-), etc. 280 V Stochastic Differential Equations In Chap. II we studied the stochastic exponential of a semimartingale. The F-S integral allows us to give a version that has a more natural appearance. Theorem 23. Let Z be a semimariingale, Zo = O. The unique solution of the equation is given by Xt=Xoexp{Zd II (1+~Zs)e-6.Zs, O 5 Fisk-Stratonovich Integrals and Differential Equations 281 The simplicity gained by using the F-S integral can be surprisingly helpful. As an example let Z = (Zl, , zn) be an n-dimensional continuous semi- martingale and let x = (Xl, ,x n)' be a column vector in JR.n. (Here' denotes transpose.) Let I be the identity n x n matrix and define xx' a(x) = 1- Ix1 2 ' (The matrix a(x) represents projection onto the hyperplane normal to x.) Note that x'a(x) = O. We want to study the system of F-S differential equa- tions Xt=xo+ lta(Xs)odZs, l~i~n. (***) where X = (Xl, ... ,Xn)' and Z = (Zl, ... , zn),. Theorem 24. The solution X of equation (* **) above exists, is unique, and it always stays on the sphere of center 0 and radius II x oII. Proof Since a is singular at the origin, we need to modify it slightly. Let g(x) be a Coo function equal to a(x) outside of a ball centered at the origin, No, such that Xo 1- No. Let Y be the solution of Yt= Xo +it g(Ys) odZs. Then Y exists and is unique by Theorem 22. Let T = inf{t > 0 : yt E No}. Since Yo = Xo and Y is continuous, P(T > 0) = 1. However for s < T, g(Ys ) = a(Ys ). Therefore it suffices to show that the function t I---' IIYtl1 is constant, since IIYo II = Ilxo II· This would imply that Y always stays on the sphere of center 0 and radius Ilxoll, and hence P(T = (0) = 1 and g(Y) = a(Y) always. To this end, let f(x) = L::I(Xi )2, where x = (x!, ... ,xn )'. Then it suffices to show that df(Yt ) = 0, (t ~ 0). Note that f is C2 • We have that for t < T, Therefore Ilytll = Ilxoll for t < T and thus by continuity T = 00 a.s. 0 Corollary. Let B = (El, ... ,En)' be n-dimensional Brownian motion, let a(x).= 1- ~i~, and let X be the solution of Then X is a Brownian motion on the sphere of radius Ilxoli. 282 V Stochastic Differential Equations Proof In Theorem 36 of Sect. 6 we show that the solution X is a diffusion. By Theorem 24 we know that X always stays on the sphere of center 0 and radius Ilxo II· It thus remains to show only that X is a rotation invariant diffusion, since this characterizes Brownian motion on a sphere. Let U be an orthogonal matrix. Then UB is again a Brownian motion, and thus it suffices to show d(UX) = a(UX) 0 d(UB). The above equation shows that UX is statistically the same diffusion as is X, and hence X is rotation invariant. The coefficient a satisfies a(Ux) = Ua(x)U', and therefore d(UX) = U 0 dX = Ua(X) 0 dB = Ua(X)U'U 0 dB = a(UX) 0 d(UB), and we are done. o The F-S integral can also be used to derive explicit formulas for solutions of stochastic differential equations in terms of solutions of (non-random) or- dinary differential equations. As an example, consider the equation Xt = Xo +I t f(Xs) 0 dZs + fat g(Xs)ds, where Z is a continuous semimartingale, Zo = 0. (Note that f; g(Xs ) 0 ds =f; g(Xs)ds, so we have not included Ito's circle in this term.) Assume that f is C2 and that f, g, and f f' are all Lipschitz continuous. Let u = u(x, z) be the unique solution of au az(x,z) = f(u(x,z)) u(x,O) = x. Then tz ~~ = f'(u(x,z))~~, and ~~(x,O) = 1, from which we conclude that ~~(x,z) = exP{faz f'(u(x,v))dv}. Let Y = (Y't)t>o be the solution of which we assume exists. For example if g~~(x, Zs)) is Lipschitz, this would ax (x, Zs) suffice. 5 Fisk-Stratonovich Integrals and Differential Equations 283 Theorem 25. With the notation and hypotheses given above, the solution X of(*4) is given by Proof. Using the F-S calculus we have Since dYs (ZS Ids =exp{- io f(u(Ys,v))dv}g(u(Ys,Zs)), we deduce By the uniqueness of the solution, we conclude that X t = u(yt, Zt). 0 We consider the special case of a simple Stratonovich equation driven by a (one dimensional) Brownian motion E. As a corollary of Theorem 25 we obtain that the simple Stratonovich equation X t = Xo +I t f(Xs) 0 dEs has a solution X t = h- 1(Bt + h(Xo)), where r 1 h(x) = io f(s) ds + C. The corresponding Ito stochastic differential equation is By explicitly solving the analogous ordinary differential equation (without using any probability theory) and composing it, we can obtain examples of stochastic differential equations with explicit solutions. This can be useful when testing simulations and numerical solution procedures. We give a few examples. Example. The equation 1 2 / 2dXt = -2a Xtdt + ay 1 - X t dBt has solution X t = sin(aBt + arcsin(Xo)). 284 V Stochastic Differential Equations Example. The following equation has only locally Lipschitz coefficients and thus can have explosions: for m::f. 1, has solution X t = (XJ-m - a(m - l)Bt ) '!=. Example. Our last example can also have explosions: has the solution X t = tan(t + B t + arctan(Xo)). The Fisk-Stratonovich integrals also have an interpretation as limits of sums, as Theorems 26 through 29 illustrate. These theorems are then useful in turn for approximating solutions of stochastic differential equations. Theorem 26. Let H be cadlag, adapted, and let X be a semimariingale. Assume [H,X] exists. Let an = {O = To :::; Tr :::; .,. :::; Tt} be a sequence of mndom partitions tending to the identity. If H and X have no jumps in common (i.e., I:o 5 Fisk-Stratonovich Integrals and Differential Equations 285 Corollary 2. Let X and Y be continuous semimartingales, and let an = {O = TO' :::; T{' :::; ... :::; TJ:;,} be a sequence of random partitions tending to the identity. Then . LIT:' T:' 1·hm -(YT :, + YT :, )(X .+' - X • ) = Ys- odXs, n-HXl 2' .+'i 0 with convergence in ucp. Proof By Theorem 22 of Chap. II, any semimartingale Y has finite quadratic variation. Thus Corollary 2 is a special case of Theorem 26 (and of Corol- lary 1). 0 Theorem 21. Let H be cadlag, adapted, of finite quadratic variation, and suppose L:o 286 V Stochastic Differential Equations For the general case we have a more complicated result. Let H be a cadlag, adapted process of finite quadratic variation. For each c > 0 we define J: = I: ~Hs1{I.:lHsl>E}. O 5 Fisk-Stratonovich Integrals and Differential Equations 287 11. Tn Tnhm L f(YTn + >'(YT!' - YTn »j,t(d>')(X '+1 - X ' )n--+oo i 0 \ '1+1 't = r f(Ys)dXs +0: r f'CYs)d[Y, X]s,JO+ JO+ with convergence in ucp. In particular if 0: = 1/2 then the limit is the F-S integral J~+ f(Ys) 0 dXs. Proof We begin by observing that 11 Tn Tn+ L j,t(d>'){f(YT,n + >'(YT:+1 - YT['» - f(YT[' )}(X '+1 - X ' ).i 0 The first sum on the right side of the above equation tends to J~+ f(Ys- )dXs in UC!p. Using the Fundamental Theorem of Calculus, the second sum on the right above equals 11 11 , Tn Tn'" j,t(d>') ds>.f (YT!' + >'S(YTn - YTn »(YT!' - YTn )(X '+1 - X ' ),L..J 't '1+1 't '/.+1 'ti 0 0 which in turn equals where F"Jw) :::; sup{If'(YT[' + >'S(YTi+1 - YTt » - f'(YTt )I}· • Since l' and Yare continuous, F"n(w) tends to 0 on compact time intervals. Also, on [0, t], . '" Tn Tn 1/2 1/2hm SUp L..,.1(YTi+1 - YT,n)(X '+1 - X ' )1 :::; [Y, Y]t [X, Xl t , n--+oo and the result follows by Theorem 30 of Chap. II. o Corollary. Let X be a semimartingale, Y be a continuous semimartingale, and f E C1. Let an = {Tt }O:Si:Skn be a sequence of random partitions tending to the identity. Then 288 V Stochastic Differential Equations with convergence in ucp. Proof Let J.l(d>.) = c{1f2}(d>'), point mass at 1/2. Then a = f01>'J.l(d>.) = 1/2, and we need only apply Theorem 29. 0 For Brownian motion, the Fisk-Stratonovich integral is sometimes defined as a limit of the form that is, the sampling times are averaged. Such an approximation does not hold in general even for continuous semimartingales (see Yor [238, page 524]), but it does hold with a supplementary hypothesis on the quadratic covariation, as Theorem 30 reveals. Theorem 30. Let X be a semimartingale and Y be a continuous semimartin- gale. Let J.l be a probability measure on [0,1] and let a = f >'J.l(d>.). Further suppose that [X, Y]t = f~ Jsds; that is, the paths of [X, Y] are absolutely con- tinuous. Let an = {tf} be a sequence of non-random partitions tending to the identity. Let f be C1. Then lim L r1f(Â¥ti+>.(ti+l_ t;J)J.l(d>.)(Xti+l - Xti) n--+oo . 10 • = r f(Ys)dXs + a r f'CYs)d[Y,X]s,10+ 10+ with convergence in ucp. In particular if a = 1/2 then the limit is the F-S integral f~+ f(Ys) 0 dXs' Proof We begin with a real analysis result. We let t; denote ti +>.(ti+1 - ti), where the ti are understood to be in an' Suppose a is continuous on [0, t]. Then which tends to O. Therefore t A tnl~~~1.' a(s)ds = >.1 a(s)ds. Moreover since continuous functions are dense in L 1([0, t], ds), the limiting result (*) holds for all a in L1([0,t],ds). 5 Fisk-Stratonovich Integrals and Differential Equations 289 Next suppose H is a continuous, adapted process, and set Then taking limits in ucp we have and using integration by parts, this equals lim{HA,n . (XY) - (HA,n X_) . Y - (HA,ny) . X}. n By Theorem 21 of Chap. II, this equals However since [X, ylt = J; Jsds, by the result (*) we conclude lim L Ht;(XtZ - Xti)(ytZ - yt,) = lim LHt;{[X, YfZ - [X, y]t,} (**) n n i i t A = li~I:Hti l' Jsds ~ , = Ait Hsd[X, y]s. We now turn our attention to the statement of the theorem. Using the Mean Value Theorem we have I:11 !(Â¥;;;)Jl(dA)(Xti+l - Xti) (* * *) • = L f(ytJ(Xti+l - Xti) + L 11 Jl(dA)f'(ytJ(Â¥;;; - ytJ(Xti+l - X t,) • • +I: r~(dA) r~s{f' (yti +S(Â¥;;A - ytJ) - f'(yti )}(Â¥;;A - ytJ(Xti+l_Xti). i Jo Jo ' , The first sum tends in ucp to J!(Ys)dXs by Theorem 21 of Chap. II. The second sum on the right side of (* * *) can be written as 290 V Stochastic Differential Equations L 11 jl(d).)j'(ytJ(yt~ - ytJ(xt~ - Xti) • + L11 jl(d).)!,(ytJ(~; - ytJ(Xti+l - xt;) . • The first sum above converges to by (**), and the second sum can be written as where Then limn KA,n . X converges to 0 locally in ucp by the Dominated Conver- gence Theorem (Theorem 32 of Chap. IV). Finally consider the third sum on the right side of (* * *). Let Fn(w) = sup sup Ij'(yti + S(ytA - ytJ) - j'(ytJI· tiECTn sE[O,1] , Then lim L r1jl (d).)Fn(w)lytA - yt.lIXti+l - xtil n i Jo ' ::::; li~11 jl(d)')Fn{L(yt~ - ytJ2}1/2{L(Xti+1 - X tY}1/2 • • =0 since lim Fn = 0 a.s. and the summations stay bounded in probability. This completes the proof. Since Y is continuous, [Y, X] = [Y, X]C, whence if a = 1/2 we obtain the Fisk-Stratonovich integral. 0 Corollary. Let X be a semimartingale, Y a continuous semimartingale, and f be C1 . Let [X, Y] be absolutely continuous and let an = {t7} be a sequence of non-random partitions tending to the identity. Then with convergence in ucp. 6 The Markov Nature of Solutions 291 Proof Let J.l(d>.) = C{1/2} (d>') , point mass at 1/2. Then J>'J.l(d>.) = 1/2, and apply Theorem 30. 0 Note that if Y is a continuous semimartingale and B is standard Brownian motion, then [Y, B] is absolutely continuous as a consequence of the Kunita- Watanabe inequality. Therefore, if f is C1 and an are partitions of [0, t], then with convergence in probability. 6 The Markov Nature of Solutions One of the original motivations for the development of the stochastic integral was to study continuous strong Markov processes (that is, diffusions), as solutions of stochastic differential equations. Let B = (Bt)t>o be a standard Brownian motion in JR.n. K. Ito studied systems of differential equations of the form X t = Xo +it f(s,Xs)dBs +it g(s, Xs)ds, and under appropriate hypotheses on the coefficients f, g he showed that a unique continuous solution exists and that it is strong Markov. Today we have semimartingale differentials, and it is therefore natural to replace dB and ds with general semimartingales and to study any resulting Markovian nature of the solution. Ifwe insist that the solution itself be Markov then the semimartingale differentials should have independent increments (see Theorem 32); but if we need only to relate the solution to a Markov process, then more general results are available. For convenience we recall here the Markov property of a stochastic pro- cess which we have already treated in Chap. 1. Assume as given a filtered probability space (fl, F, (Ftk:~o,P) satisfying the usual hypotheses.6 Definition. A process Z with values in JR.d and adapted to IF = (Fth:::::o is a simple Markov process with respect to IF if for each t ~ 0 the a-fields Ft and a{Zu;u ~ t} are conditionally independent given Zt. Thus one can think of the Markov property as a weakening of the property of independent increments. It is easy to see that the simple Markov property is equivalent to the following. For u ~ t and for every f bounded, Borel measurable, One thinks of this as "the best prediction of the future given the past and the present is the same as the best prediction of the future given the present." 6 See Chap. I, Sect. 1 for a definition of the "usual hypotheses" (page 3). 292 V Stochastic Differential Equations Using the equivalent relation (*), one can define a transition function for a Markov process as follows, for s < t and f bounded, Borel measurable, let Ps,t(Zs,J) = E{f(Zt)IFs}' Note that if f (x) = 1A (x), the indicator function of a set A, then the preceding equality reduces to P(Zt E AIFs) = Ps,t(Zs, IA). Identifying lA with A, we often write Ps,t(Zs, A) on the right side above. When we speak of a Markov process without specifying the filtration of a- algebras (Ftk:~o, we mean implicitly that :Pi = a{Zs; s ::::; t}, the natural filtration generated by the process. It often happens that the transition function satisfies the relationship Ps,t = Pt- s for t ~ s. In this case we say the Markov process is time homogeneous, and the transition functions are a semigroup of operators, known as the transition semigroup (Ptk::~o. In the time homogeneous case, the Markov property becomes P(Zt+s E AIFt ) = Ps(Zt, A). A stronger requirement that is often satisfied is that the Markov property hold for stopping times. Definition. A time homogeneous simple Markov process is strong Markov if for any stopping time T with P(T < (0) = 1, s ~ 0, P(ZT+s E AIFT) = Ps(ZT,A) or equivalently E{f(ZT+s) 1FT} = Ps(ZT, f), for any bounded, Borel measurable function f. The fact that we defined the strong Markov property only for time ho- mogeneous processes is not much of a restriction, since if X is an JR.d-valued simple Markov process, then it is easy to see that the process Zt = (Xt , t) is an JR.d+l-valued time homogeneous simple Markov process. Examples of strong Markov processes (with respect to their natural fil- trations of a-algebras) are Brownian motion, the Poisson process, and indeed any Levy process by Theorem 32 of Chap. 1. The results of this section will give many more examples as the solutions of stochastic differential equations. Since we have defined strong Markov processes for time homogeneous pro- cesses only, it is convenient to take the coefficients of our equations to be autonomous. We could let them be non-autonomous, however, and then with an extra argument we can conclude that if X is the solution then the process yt = (Xt , t) is strong Markov. We recall a definition from Sect. 3 of this chapter. 6 The Markov Nature of Solutions 293 Definition. A function J : JR.+ x JR.n ----> JR. is said to be Lipschitz if there exists a finite constant k such that (i) IJ(t, x) - J(t, y)1 ::; klx - YI, each t E JR.+, and (ii) t I---' J(t, x) is right continuous with left limits, each x E JR.n. J is said to be autonomous if J(t,x) = f(x). In order to allow arbitrary initial conditions, we need (in general) a larger probability space than the one on which Z is defined. We therefore define n = JR.n X n -=t F~. A random variable Z defined on n is considered to be extended automatically to n by the rule Z(w) = Z(w), when w = (Y,w). We begin with a measurability result which is an easy consequence of Sect. 3 of Chap. IV. Theorem 31. Let zj be semimartingales (1 ::::; j ::::; d), H X a vector oj adapted processes in lI)) for each x E JR.n , and suppose (x, t, w) I---' Ht (w) is B ® B+ ® F measurable.7 Let FJ be functional Lipschitz and for each x E JR.n , Xx is the unique solution oj There exists a version of xx such that (x,t,w) I---' Xt(w) is B ® B+ ® F measurable, and for each x, Xt is a cadlag solution of the equation. Proof Let Xo(x, t,w) = Ht(w) and define inductively d t X n+1 (x, t,W)i = HtX +L 1FJ(Xn(x,., ·»s-dz1· j=l ° The integrands above are in L, hence by Theorem 63 in Chap. IV there exists measurable, cadlag versions of the stochastic integrals. By Theorem 8 the processes xn converge ucp to the solution X for each x. Then an application of Theorem 62 of Chap. IV yields the result. 0 7 B denotes the Borel sets on lR.n ; B+ the Borel sets on lR.+. 294 V Stochastic Differential Equations We state and prove the next theorem for one equation. An analogous result (with a perfectly analogous proof) holds for finite systems of equations. Theorem 32. Let Z = (Zl, ... , Zd) be a vector of independent Levy pro- cesses, Zo = 0, and let (IJ) 1 ::; j ::; d, 1 ::; i ::; n, be Lipschitz functions. Let Xo be as in (*) and let X be the solution of Then X is a Markov process, under each p Y and X is strong Markov if the fj are autonomous. Proof We treat only the case n = 1. Let T be an IF stopping time, T < 00 a.s. Define gT = a{ Z?+u - Z?; u ~ 0,1 ::; j ::; d}. Then gT is independent of FT under pY, since the Zj are Levy processes, as a consequence of Theorem 32 of Chap. I. Choose a stopping time T < 00 a.s. and let it be fixed. For u ~ 0 define inductively yO(x, T, u) = X, d IT+u yn+l(x,T,u) =x+ L fj(v-,yn(x,T,v-»dZt· j=l T Also, let X (x, T, u) denote the unique solution of d rT +u X(x,T,u) = x +I: J7 fj(v-,X(x,T,v-»dZ~, j=l T taking the jointly measurable version (cf., Theorem 31). By Theorem 8 we know that X (x, T, u) is gT measurable. By approximating the stochastic integral as a limit of sums, we see by induction that yn (x, T, u) is gT measurable as well. Under pX we have X(Xo,T,u) = X(x,T,u) a.s., and yn(xo, T, u) = yn(x, T, u) pX_a.s., also. By uniqueness of solutions and us- ing Theorem 31, for all u ~ 0 a.s. X(Xo,0, T + u) = X(X(Xo,0, T), T, u). There is no problem with sets of probability zero, due to (for example) the continuity of the flows. (See Theorem 37.) Writing EX to denote expectation on n with respect to pX, and using the independence of FT and gT (as well as of FT and '{F), we have for any bounded, Borel function h EX{h(X(Xo,0, T + u»IFT} = E{h(X(x, 0, T + u»IFT}lIR = E{h(X(X(x"O,T),T,u»}lIR = j(X(x, 0, T»l IR , 6 The Markov Nature of Solutions 295 where j(y) = E{h(X(y, T, un. The last equality follows from the elementary fact that E{F(H, ')IH} = f(H), where f(h) = E{F(h, ·n, if F is independent of Hand H is H measurable. This completes the proof, since the fact that EX{h(X(Xo,0, T + u»)IFT} is a function only of X(x, 0, T) implies that EX{h(X(Xo,0, T + u)IFT} = EX{h(X(Xo,O, T + u»)IX(Xo,O, Tn. 0 It is interesting to note that Theorem 32 remains true with Fisk-Straton- ovich differentials. To see this we need a preliminary result. Theorem 33. Let Z = (Zl, ... , Zd) be a vector of independent Levy pro- cesses, Zo = O. Then [Zi,Zj]c = ° ifi i:- j, and [Zi,Zi]r = at, where a = E{[Zi, Zi]i}. Proof First assume that the jumps of each Zi are bounded. Then the mo- ments of Zi of all orders exist (Theorem 34 of Chap. I), and in particular Mf == zf - E{Zi} is an L2 martingale for each i, with E{Zi} = tE{Zn. By independence Mi Mj is also a martingale and hence [Mi, Mj] = °by Corol- lary 2 of Theorem 27 of Chap. II. Therefore [Zi, Zj]r = [Mi , Mj]r = °as well. Next consider A~ == [Zi, Zi]t = [Mi, Milt- It is an immediate consequence of approximation by sums (Theorem 22 of Chap. II) that Ai also has inde- pendent increments. Since A~ = [Mi,Mi]r + L (~M;)2, O 296 V Stochastic Differential Equations d t XI = X6 +L 1fJ(s-, X s-) 0 dZ{ j=1 0 Then X is a Markov process and X is strong Markov if the coefficients fj are autonomous. Proof. We treat only the case n = 1. By Theorem 33 [Zi, Zj]i = 0 if i i:- j and [Zi, Zi]f = ait, ai :::: o. Therefore, by Theorem 22 the equation is equivalent to and since the process yt = t is a Levy process we need only to invoke Theo- rem 32 to complete the proof. 0 If the differentials are not Levy processes but only strong Markov pro- cesses which are semimartingales, then the solutions of equations such as (**) in Theorem 32 need not be Markov processes. One does, however, have the following result. Theorem 35. Let Z be a strong Markov processes with values in JR., Zo = 0, such that Z is a semimartingale. Let f and g be Lipschitz functions. Let X o be as in Theorem 32 and X be the solution of t t X t = Xo +1f(s-,Xs-)dZs + io g(s,Xs_)ds. Then the vector process (X, Z) is Markov under P, each Y E JR., and strong Markov if f and g are autonomous. Proof First recall that X is defined on n = JR. x n and Z is automatically extended to n as explained at the beginning of this section. The probability -y -yp = Cy x P is such that P (Xo = y) = 1, each Y E JR.. As in the proof of Theorem 32 for a fixed stopping time T (T < 00 a.s.) and x E JR. define inductively for u :::: 0 yO(x, T, u) = x rT+uyn+1(x, T, u) = x + iT f(v-, yn(x, T, v-»dZv rT+u + iT g(v-,yn(x,T,v-»dv, and let X(x, T, u) denote the unique solution of rT+U rT+uX(x,T,u) =x+ iT f(v-,X(x,T,v-»dZv + iT g(v-,X(x,T,v-»dv. 6 The Markov Nature of Solutions 297 Next define gT = a{ZT+u - ZT;U ~ O} for the same stopping time T. As in the proof of Theorem 32 we see that yn(x, T, u) is gT measurable for each n ~ 1 and that X(x, T, u) is gT measurable as well. Next let h be Borel measurable and bounded, and let G E gT and bounded. Observe that E{h(XT)GIFT } = h(XT)E{GIFr} = h(XT)E{GIZT} = j(XT,ZT), where j : JR2 --t JR is Borel. Since XT+u = X(XT,T, u) by the uniqueness of the solution, the theorem now follows from the Monotone Class Theorem (Theorem 8 of Chap. I). 0 Theorem 35 can also be shown to be true with Stratonovich differentials, but the proof is more complicated since the quadratic variation process is an additive functional of Z, rather than a deterministic process (as is the case when Z is a Levy process). For this type of result we refer the reader to Qinlar-Jacod-Protter-Sharpe [34]. Each of Theorems 32, 34 and 35 can be interpreted with X having an arbitrary initial distribution /-l. Indeed for /-l a probability measure on (JR, B), -I-' - -I-' -xdefine P on 0 by P (A) = fn~ P (A)/-l(dx), for A E B 0 F. Then the conclu- sions of the three theorems are trivially still valid for pI-', and the distribution of X o is /-l. Traditionally the most important Markovian solutions of stochastic dif- ferential equations are diffusions. Suppose as given a space (0, F, (Ft}t~O,P) satisfying the usual hypotheses. Definition. An adapted process X with values in JRn is a diffusion9 if it has continuous sample paths and if it satisfies the strong Markov property. A restatement of Theorems 32 and 34 yields the following. Theorem 36. Let B = (B 1 , ... , Bd) be a standard Brownian motion on JRd, Bo = 0, and let (ff) 1 :::; j :::; d, 1 :::; i :::; n, gi be autonomous Lipschitz functions. Let Xb be as in Theorem 32 and let X be the solution of Then X is a diffusion. If the coefficients (fJh-:;j -:;d, 1 :::; i :::; n, are non-random F-S acceptable functions and if Y is the solution of d t t~i = yoi + L 1fJ(ys) 0 dB~ + rgi(Ys)ds, j=1 0 io 9 The definition of a diffusion is not yet standardized. We give a general definition. 298 V Stochastic Differential Equations then Y is a diffusion. In equation (* **) of Theorem 36 the coefficients Ij are called the diffusion coefficients and the coefficients gi are called the drift coefficients. Note that a diffusion need not be semimartingale, even though of course the solutions of equations such as (* * *) are semimartingales. Indeed any deterministic continuous function with paths of unbounded variation is a dif- fusion which is not a semimartingale. Another interesting example is provided by Tanaka's formula. The process is a diffusion and it is a semimartingale, where B is standard Brownian motion on R Since the paths of L~ are singular with respect to Lebesgue measure, and since a semimartingale decomposition is unique for continuous processes (the corollary of Theorem 31 of Chap. III), we see that IBtl cannot be represented as a solution of (* * *). Another example is that of IBt 11/3 which is a time homogeneous diffusion but not a semimartingale (Theorem 71 of Chap. IV). An intuitive notion of a diffusion is to imagine a pollen grain floating down- stream in a river. The grain is subject to two forces: the current of the river (drift), and the aggregate bombardment of the grain by the surrounding wa- ter molecules (diffusion). The coefficient I(t,x) then represents the sensitivity of the particle at time t and place x to the diffusion forces. For example, if part of the river water is warmer at certain times and places (due to sunlight or industrial effluents, for example), then I might be larger. Analogously g would be larger when the river was flowing faster due to a steeper incline. We give three simple examples of diffusions. Example. The stochastic exponential exp{Bt - ~t} is a diffusion where f(t,x) = x and g(t,x) = o. Example. Consider the simple system lit = Vo +it adBs +it a~ds, X t = X o+l t Vsds. The process X can be used as a model of Brownian motion alternative to Einstein's. It is called the Ornstein-Uhlenbeck Brownian motion, or simply the Ornstein-Uhlenbeck process. Note that here the process X has paths of finite variation and hence the process (lit k::o is a true velocity process for X. Using integration by parts we can verify that 6 The Markov Nature of Solutions 299 is an explicit solution for V. Indeed e-atvt = Vo + it Vs(-o:e-aS)ds + it e-asdv" = Vo - 0: it v"e-asds + it e-aso:v"ds + it e-aSadBs = Vo +I t e-asadBs, and we are done. Since Vo and the Brownian motion B are independent (by construction), we see that when Vo has a Gaussian distribution then V is a Gaussian process. If 0: is negative and we take a 2 = -1/(20:), then V is a stationary Gaussian process. Example. Consider next the equation I t -XsX t = --ds + B t ,o 1 - s (0 ~ t < 1) for B a standard Brownian motion. For each to, 0< to < 1, there is a solution which is a diffusion on [0, to]. By the uniqueness of solutions if tl < to then the solution for [0, to] agrees with the solution for [0, tl] on the interval [0, tIl. Thus we have a solution on [0,1). If we can show that limt--->l X t = 0, then the solution extends by continuity to [0,1] and we will have constructed a diffusion X on [0,1] with X o = Xl = 0, known as the Brownian bridge. lO An application of integration by parts shows that the solution of the Brow- nian bridge equation is given by i t 1X t = (1 - t) -1- dBso - s (0 ~ t < 1). Write X t = f(t)Mt , where f(t) = (l-t) and M t = f~(I-s)-ldBs' To see that limt---> I X t = 0, we study M t , making the change of variables t = u(1 + u)-l. Then °~ t < 1 corresponds to °~ u < 00, and define N u = M~ Qu = :F,+u u , °~ U < 00.l+u' N is clearly a continuous Qu martingale, and moreover [N, N]u = u; hence N is a standard Brownian motion by Levy's Theorem (Theorem 39 of Chap. II). It is then easy to show, as a consequence of the Strong Law of Large Numbers, that11 . Nt hm - = ° a.s.t--->oo t 10 The Brownian bridge is also known as tied down Brownian motion, and alterna- tively as pinned Brownian motion. 11 See, for example, Breiman [23, page 265]. 300 V Stochastic Differential Equations Let g(t, w) = Nt(w)jt so that limt--->o g(C l , w) = 0 a.s. Thus, limt--->o tNl/ t = 0 a.s. We then have, replacing t with (1 - t) lim(1 - t)Nl/(l-t) = lim(1 - t)Ml /(2-t) = 0 a.s.,t--->l t--->l and therefore limu --->l(l - u)Mu = 0 a.s., and hence limt--->l X t = 0 a.s. We now know that X is a continuous diffusion on [0, 1], and that X o = Xl = 0 a.s. Also X is clearly a semimartingale on [0,1), but it is not obvious that X is a semimartingale on [0, 1]. One needs to show that the integral fol ~ds < 00 a.s. To see this calculate E{Xn, 0 :::; t < 1, By the Cauchy-Schwarz inequality E{/Xtl} :::; E{X;}1/2 = Jt(l - t). Therefore E{ rl /Xsi ds} = rl E{IXs/} dsJo 1 - s Jo 1 - s < rl Js(l-s)ds 7 Flows of SDE: Continuity and Differentiability 301 7 Flows of Stochastic Differential Equations: Continuity and Differentiability Consider a stochastic differential equation of the form Obviously there is a dependence on the initial condition, and we can write the solution in the form X(t, w, x), or Xf(w). The study of the flow of a stochastic differential equation is the study of the functions ¢ : x -. X (t, w, x) which can be considered as mapping lRn -. lRn for (t, w) fixed, or as mapping lRn -. V n , where vn denotes the space of cadla,g functions from lR+ to lRn , equipped with the topology of uniform convergence on compacts. It is important to distin- guish between V n and J1))n. The former is a function space, and it is associated in the literature with weak convergence results (see, e.g., Billingsley [17], or Ethier-Kurtz [71], or Kurtz-Protter [136] for recent results in a semimartin- gale context); the latter is the space of stochastic processes with dtdlag paths, and which are adapted to the underlying filtration. Note that we have already encountered flows in Sect. 6 (d., Theorem 31), where we proved measurability of the solution with respect to a parameter (which can be taken to be, of course, the initial condition). We will be inter- ested in several properties of the flows: continuity, differentiability, injectivity, and when the flows are diffeomorphisms of lRn . We begin with continuity. We consider a general system of equations of the form where Xf and Hf are column vectors in lRn , Z is a column vector of m semimartingales with Zo = 0, and F is an n x m matrix with elements (F~). For x fixed, for each y we have that X t = Xr - Xf is a solution of the equation X t = Hi - H: + it F(X)s-dZs, where F(Y) = F(XX + Y) - F(XX). Theorem 37. Let HX be processes in J1))n, and let x f-+ HX : lRn -. J1))n be prelocally Lipschitz continuous in .tiP, some p > n. Let F be an n x m matrix of functional Lipschitz operators (F~), 1 ::; i ::; n, 1 ::; a ::; m. 12 Then there exists a function X (t, w, x) on lR+ x n x lRn such that (i) for each x the process XtX(w) = X(t,w,x) is a solution of (*), and (ii) for almost all w, the flow x f-+ X (-, w, x) from lRn into V n is continuous in the topology of uniform convergence on compacts. 12 See page 250 for the definition of functional Lipschitz. 302 V Stochastic Differential Equations Proo]. We recall the method of proof used to show the existence and unique- ness of a solution (Theorem 7). By stopping at a fixed time to, we can assume the Lipschitz process is just a random variable K which is finite a.s. Then by conditioning13 we can assume without loss of generality that this Lipschitz constant is non-random, and we call it c < 00. By replac- ing Hf with Xf + J~ F(O)s_dZs, and then by replacing F with G given by G(y}t = F(Y)t - F(O)t, we can further assume without loss of generality that F(O) = O. Then for (3 = Cp(c), by Theorem 5 we can find an arbitrarily large stopping time T such that ZT- E S({3), and HX is Lipschitz continuous in gP on [0, T). Then by Lemma 2 (preceding Theorem 7) we have that for the solution X of (**) and some (finite) constant Cp(c, Z). Choose p > n, and we have E{sup IX: - XfIP } :::; Cp(c, Z)Kllx - yllP, s 7 Flows of SDE: Continuity and Differentiability 303 Definition. A function f : JRn -+ JR is said to be locally Lipschitz if there exists an increasing sequence of open sets Ak such that Uk Ak = JRn and f is Lipschitz with a constant K k on each Ak . For example, if f has continuous first partial derivatives, then it is locally Lipschitz, while if its continuous first partials are bounded, then it is Lipschitz. If f, g, are both Lipschitz, then their product fg is locally Lipschitz. These coefficients arise naturally in the study of Fisk-Stratonovich equations (see Sect. 5). Theorem 38. Let Z be as in (Hi) and let the functions (f~) in (H2) be locally Lipschitz. Then there exists a function ((x, w) : JRn X n -+ [O,ooJ such that for each x ((x,·) is a stopping time, and there exists a unique solution of (00) up to ((x,·) with limsuPt--->«(x,) IIXtl1 = 00 a.s. on {( < oo}. Moreover x I--t ((x,w) is lower semi-continuous, strictly positive, and the flow of X is continuous on [0, ((x, .)). Remark. Before proving the theorem we comment that for each x fixed the stopping time T(w) = ((x,w) is called an explosion time. Thus Theo- rem 38 assures the existence and uniqueness of a solution up to an explosion time; and at that time, the solution does indeed explode in the sense that lim SUPt--->T IIXt II = +00 on {T < oo}. Note however that if the coefficients (f~) in (H2) are (globally) Lipschitz, then a.s. ( = 00 for all x (Theorem 7). Proof Let At be open sets increasing to JRn such that there exist (ht), a sequence of Coo functions with compact support mapping JRn to [0,1] such that ht = Ion At. For each f. we let Xt(t, w, x) denote the solution (continuous in x) of the equation Xt= x + It gt(Xs - )dZs where the matrix gt(x) is defined by ht(x)f(x). Define stopping times, for x fixed, St(w,x) = inf{t > 0: Xt(t,w,x) tJ- At or Xt(t-,w,x) tJ- At}. By Theorem 37 the flow is uniformly continuous on compact sets, hence the functions St are lower semi-continuous. Then on [0, St) we have that X t = X t +1 by the uniqueness of the solutions, since they both satisfy equation (00). We wish to show that the relation (L) holds for all x E JRn simultaneously. Choose an x and let 304 V Stochastic Differential Equations Then P(AX) = 1. We set A = UXEiQln AX, where iQn denotes the rationals in lRn . Then P(A) = 1 as well, and without loss of generality we take A = f2. Next for arbitrary y E lRn, let Yn -+ y, with Yn E iQn. Then and therefore the relation (L) holds for Y as well, by the previously established continuity. Observe that St(w,x) :::; S£+l(W,X), and let ((x,w) = SUPtSt(w,x), for each x. Then ( is lower semi-continuous because the St are, and it is strictly positive. Further, we have shown that there exists a unique func- tion X(·,w,x) on [O,((x,w)) which is a solution of (00), and it is equal to Xe(-,w,x) on [O,Se(w,x)), each £ ~ 1. Indeed, on [O,Se(w,x)) we have X(-,w, x) = X e(·, w, x), and since X e(-, w, x) E Ae, we have he(Xe("w, x)) = 1, wherefore X t is a solution of (00) on [0, Se). By our construction we have that X(St(w,x),w,x) belongs to Aj. Indeed, letting St(w, x) = s, then X(s-, w, x) = Xt(s-, W, x) = u E At, for some value u. Since u E At we have ht(u) = 1, and thus X and X t have the same jump at S and we can conclude X(s,w, x) = Xt(s,w, x). But Xt(s, w,x) or Xt(s-, w, x) must be in Ai by right continuity and the definition of St; therefore X(s,w,x) or X(s-,w, x) is in Aj. This shows that limsuPt-->( IIXtl1 = 00 on {( < oo}. 0 It is tempting to conclude that if there are no explosions at a finite time for each initial condition x a.s. (null set depending on x), then also ((x, .) = 00 a.s. (with the same null set for all initial values x). Unfortunately this is not true as the next example shows. Example. Let B be a complex Brownian motion. That is, let Bl and B2 be two independent Brownian motions on lR and let Bt == Bi + iB;, where i 2 = -1. Consider the stochastic differential equation Zt = z -it Z;dBs, where the initial value z is complex. This equation has a closed form solution given by z Z(t, w, z) = 1 + zBt(w) Indeed, if we set f(x) = z(1 + ZX)-l and Zt = f(Bd, then f'(Bd = -Z;, and since B t = BI + iB'f, we have by Ito's formula since d[Bl, Bl]s = d[B2, B 2]s = ds, we see that Zt = z - f~ Z;dBs. 7 Flows of SDE: Continuity and Differentiability 305 For z fixed we know that P(3 t : Bt = -1/z) = 0, since Brownian motion in ]R2 a.S. does not hit a specified point. Therefore Z does not have explosions in a finite time for each fixed initial condition Zo = z. On the other hand for any Wo and to fixed we have B ta (wo) = zo for some Zo E C, and thus for the initial value z = -1/Zo we have an explosion at the chosen finite time to. Thus each trajectory has initial conditions such that it will explode at a finite time, and P{w: 3z with (w,z) < oo} = l. We next turn our attention to the differentiability of the flows. To this end we consider the system of n+n2 equations (assuming that the coefficients (f~) are at least C1 ) xi = Xi + f it f~(Xs-)dZ': ,,=1 0 D kt = 15k+ f tit ~~~(Xs-)Dks_dZ':, ,,=1 j=1 0 J (D) (D) (1 :::; i :::; n) where D denotes an n x n matrix-valued process and 8~ = 1 if i = k and 0 otherwise (Kronecker's delta). A convenient convention, sometimes called the Einstein convention, is to leave the summations implicit. Thus the system of equations (D) can be alternatively written as X: = Xi + It f~(Xs- )dZ': t i D i - ,i 1&f" (X )Dj dZ"kt - Uk + -- s- ks- s' o aXj We will use the Einstein convention when there is no ambiguity. Note that in equations (D) if X is already known, then the second system is linear in D. Also note that the coefficients for the system (D) are not globally Lipschitz, but if the first partials of the (f~) are locally Lipschitz, then so also are the coefficients of (D). Theorem 39. Let Z be as in (Hi) and let the functions (f~) in (H2) have locally Lipschitz first partial derivatives. Then for almost all w there exists a function X(t,w,x) which is continuously differentiable in the open set {x : «x,w) > t}, where ( is the explosion time (cf., Theorem 38). If (f~) are globally Lipschitz then (= 00. Let Dk(t,w,x) == f)~kX(t,W,X). Then for each x the process (X(·,w,x),D(·,w,x)) is identically cadlag, and it is the solution of equations (D) on [0, (x, .)). Proof. We will give the proof in several steps. In Step 1 we will reduce the problem to one where the coefficients are globally Lipschitz. We then resolve the first system (for X) of (D), and in Step 2 we will show that, given X, there exists a "nice" solution D of the second system of equations, which depends 306 V Stochastic Differential Equations continuously on x. In Step 3 we will show that Dt is the partial derivative in Xk of Xi in the distributional sense.14 Then since it is continuous (in x), we can conclude that it is the true partial derivative. Step 1. Choose a constant N. Then the open set {x : ((x,w) > N} is a countable union of closed balls. Therefore it suffices to show that if B is one of these balls, then on the set r = {w : \:Ix E B, ((x, w) > N} the function x I--t X (t, w, x) is continuously differentiable on B. However by Theorem 38 we know that for each wE r, the image of X as x runs through B is compact in V n with 0 :::; t :::; N, hence it is contained in a ball of radius R in lRn , for R sufficiently large. We fix the radius R and we denote by K the ball of radius R of lRn centered at O. Let A = {w: for x E B and 0 :::; t :::; N, X(t, w, x) E K}. We then condition on A. That is, we replace P by PA, where PA(A) == P(AIA) = P~1~~)· Then PA « P, so Z is still a semimartingale with re- spect to PA (Theorem 2 of Chap. II). This allows us to make, without loss of generality, the following simplifying assumption: if x E B then ((x, w) > N, and X(t,w,x) E K, 0:::; t:::; N. Next let h : lRn --t lR be Coo with compact support and such that h(x) = 1 if x E K and replace f with fh. Let Z be implicitly stopped at the (constant stopping) time N. (That is, ZN replaces Z.) With these assumptions and letting PA replace P, we can therefore assume-without loss of generality- that the coefficients in the first equation in (D) are globally Lipschitz and bounded. Step 2. In this step we assume that the simplifying assumptions of Step 1 hold. We may also assume by Theorem 5 that Z E S({3) for a {3 which will be specified later. If we were to proceed to calculate formally the derivative with respect to Xk of Xi, we would get Therefore our best candidate for the partial derivative with respect to Xk is the solution of the system 14 These derivatives are also known as derivatives in the generalized function sense. which in turn implies 7 Flows of SDE: Continuity and Differentiability 307 and let D be the matrix (Di). Naturally X s = X(s,w,x), and we can make explicit this dependence on x by rewriting the above equation as Dix = 6i +it af~ (XX )DjX dZakt k ;::, s- ks- s , o uXj where the summations over a and j are implicit (Einstein convention). We now show that Dk = (D~, ... , D;:) is continuous in x. Fix x, y E IRn and let ~(w) = Dk(S,W,X) - Dk(s,w,y). Then . Tn fS afi where Y}: = La=l Jo ~(X~_)dZ::. Note that by Step 1 we know that ~(X~_) is bounded; therefore since za E S((3), the Yji are in S(c(3) for a constant c. If (3 is small enough, by Lemma 2 (preceding Theorem 7) we have that for each p 2: 2 there exists a constant Op (Z) such that However we can also estimate IIHIlgP. If we let Ji =~ {Of~ (XX) _ of~ (XY )} Djx as L..J ax. s- ax. s- ks-'j=l J J then Hi = L:'=l J~ J~sdZ'i, and therefore by Emery's inequality (Theorem 3) we have that 1[VllgP :S Cp(Z)IIJllgp. We turn our attention to estimating II JII sp. Consider first the terms 308 V Stochastic Differential Equations By the simplifying assumptions of Step 1, the functions ~ are Lipschitz in J K, and Xx takes its values in K. Therefore Theorem 37 applies, and as we saw in its proof (inequality (* * *)) we have that Next consider the terms D~~. We have seen that these terms are solutions of the system of equations and therefore they can be written as solutions of the exponential system Djx = oj + it D ex dy'jXkt k ks- is' o with Yl,x = J; ~(X~_)dZ~. As before, by Lemma 2, Recalling the definition of J and using the Cauchy-Schwarz inequality gives IIJllgp:S II~~:(X:_) - ~~:(X%_)llg2PIID~Xllg2P :S C2p (Z)llx - yll, which in turn combined with previous estimates yields 1IVIIsp :S C(p,Z)llx - yll· Since V was defined to be Vs(w) = Dk(s,w,x) - Dk(S,w,y), we have shown that (with p > n) E {sup IIDk(S, w, x) - Dk(S, w, y)IIP} :S C(p, Z)Pllx - ylIP, s~N and therefore by Kolmogorov's Lemma (Theorem 72 of Chap. IV) we have the continuity in x of Dk(t,w,x). Step 3. In this step we first show that Dk(t,w,x), the solution of equations (*4) (and hence also the solution of the n 2 equations of the second line of (D)) is the partial derivative of X in the variable Xk in the sense of distributions (Le., generalized functions). Since in Step 2 we established the continuity of Dk in x, we can conclude that Dk is the true partial derivative. 7 Flows of SDE: Continuity and Differentiability 309 Let us first note that with the continuity established in Step 2, by increas- ing the compact ball K, we can assume further that Dk(S,W,X) E K also, for S :S N and all x E B. We now make use of Cauchy's method of approximating solutions of dif- ferential equations, established for stochastic differential equations in The- orem 16 and its corollary. Note that by our simplifying assumptions, Y = (X, D) takes its values in a compact set, and therefore the coefficients are (globally) Lipschitz. The process Y is the solution of a stochastic differential equation, which we write in vector and matrix form as Let U r be a sequence of partitions tending to the identity, and with the no- tation of Theorem 16 let Y(u) = (X(u),D(u)) denote the solution of the equation of the form yt = y +it f(ya+)~dZs' For each (ur ) the equations (D) become difference equations, and thus trivially The proof of the theorem will be finished if we can find a subsequence rq such that limq -+oo X(urq ) = X and limq -+oo Dk(Urq ) = Dk, in the sense of distributions, considered as functions of x. We now enlarge our space n exactly as in Sect. 6 (immediately preceding Theorem 31): n = 1R2n X n F: = B2n @Ft , F t = nF:, u>t where B2n denotes the Borel sets of 1R2n . Let>. be normalized Lebesgue mea- sure of K. Finally define P=>'xp. We can as in the proof of Step 2 assume that Z E 8((3) for (3 small enough and then, lim (X(ur ), D(ur ) = (X, D) in s.?, r--+OO - by Theorem 16. Therefore there exists a subsequence rq such that 00 M = LsuPII(X(rq),Dh)) - (X,D)II E L 1 (dP). q=l t 310 V Stochastic Differential Equations The function M = M(w, x) is in L 1 (>. x P), and therefore for P-almost all w the function x f---t M(w, x) E L 1 (d>.). For w not in the exceptional set, and t fixed it follows that lim (X(rq),D(rq)) = (X,D) q--->oo >. a.e. Further, it is bounded by the function M(w,·) +II(X(t, w, '), D(t, w, '))11 which is integrable by hypothesis. This gives convergence in the distributional sense, and the proof is complete. D We state the following corollary to Theorem 39 as a theorem. Theorem 40. Let Z be as in (Hi) and let the functions (f~) in (H2) have locally Lipschitz derivatives up to order N, for some N, 0 :::; N :::; 00. Then there exists a solution X(t, w, x) to which is N times continuously differentiable in the open set {x: ((x, w) > t}, where ( is the explosion time of the solution. If the coefficients (f~) are globally Lipschitz, then ( = 00. Proof. If N = 0, then Theorem 40 is exactly Theorem 38. If N = 1, then Theorem 40 is Theorem 39. If N > 1, then the coefficients of equations (D) have locally Lipschitz derivatives of order N - 1 at least. Induction yields (X,D) E eN-I, whence X E eN. D Note that the coefficients (f~) in Theorem 40 are locally Lipschitz of order N if, for example, they have N + 1 continuous partial derivatives; that is, if f~ E eN +l(lRn ), for each i and a, then (f~) are locally Lipschitz of order N. 8 Flows as Diffeomorphisms: The Continuous Case In this section we will study a system of differential equations of the form X: = Xi + f, t F~(X)s_dZ':, 1:::; i:::; n, ",=1 io where the semimartingales Z'" are assumed to have continuous paths with Zo = O. The continuity assumption leads to pleasing results. In Sect. 10 we consider the general case where the semimartingale differentials can have jumps. The flow of an equation such as (*) is considered to be an IRn -valued function 8 Flows as Diffeomorphisms: The Continuous Case 311 Definition. The flow 312 V Stochastic Differential Equations times Sk strictly less than T. Therefore the process l[O,T) = limk-+oo l[o,skJ is predictable. We use Ito's formula (Theorem 32 of Chap. II) on [0, T) for the function f(x) = log Ilxll. Note that of 1 ollxll 1 xi --- ---- OXi Ilxll OXi Ilxllllxll 02 f 1 2Xi ollxll ox; = IIxl1 2 - IIxll3 OXi 0 2 f 2XiXj OXiOXj -W· Therefore on [0, T), IIx1/ 2 ' 1 2x2 -----, . IIxl1 2 Ilxl/4' log IJUt II - log II u II t t t i . _ " flUid i 1" f 1 d[ i Ui] "f UsUl d[ i j] - L..-Jo IIUsl12 s Us + 2" L..-Jo IIUsl12 U, s - L.-JJo IlUsl14 U ,U s· I I I,) Since dUi = E", Vi''''dZ''', the foregoing equals All the integrands on the right side are predictable and since IIVII ::::; KIIUII they are moreover bounded by K and K 2 in absolute value. However on {T < oo} the left side of the equation, log IlUtli -log Ilull, tends to -00 as t increases to T; the right side is a well-defined non-exploding semimartingale on all of [0,00). Therefore P(T < 00) = 0, and the proof is complete. 0 In the study of strong injectivity the stochastic exponential of a semi- martingale (introduced in Theorem 37 of Chap. II) plays an important role. Recall that if Z is a continuous semimartingale, then Xo£(Z) denotes the (unique) solution of the equation X t = X o+I t XsdZs, and £(Z)t = exp{Zt - ![Z, Z]t}. In particular, P(infs~t£(Z)s > 0) = 1. Theorem 42. For x E IRn , let HX be in JI)k such that they are locally bounded uniformly in x. Assume further that there exists a sequence of stopping times (T£k~l increasing to 00 a.s. such that II(H: - H~)Tillgr ::::; Kllx - yll, each 8 Flows as Diffeomorphisms: The Continuous Case 313 £ 2 1, for a constant K and for some r > n. Let Z = (Zl, ... , Zk) be k semimartingales. Then the functions x I t H:dZs x [HX • Z, H X • Z]t have versions which are continuous as junctions from lRn into V, with V having the topology of uniform convergence on compacts. Proof. By Theorem 5 there exists an arbitrarily large stopping time T such that ZT- E Hoc. Thus without loss of generality we can assume that Z E Hoc, and that HXis bounded by some constant K, uniformly in x. Furthe-;-we assume IIH: - H~IIEr :S Kllx - yll. Then E{sup II t H:_dZs - t H;_dZsln :S CE{sup IIH;_ - Hi-lnllZlllioo t Jo Jo t - :S Kllx - YlnlZlllioo, where we have used Emery's inequality (Theorem 3). The result for JH;_dZs now follows from Kolmogorov's Lemma (Theorem 72 of Chap. IV). For the second result we have II[H:· Z,H:· Z] - [H~. Z,H~· Z]lIg r = II J{(H:_)2 - (Hf_)2}d[Z, Z]sllgr k = II L J(H;~) + H%~(H:~ - H%~)d[Zi,Zj]sllgr, i,j=l :S 2KIIZIlkoo IIHx - HYlIsr, - - and the result follows. o Theorem 43. Let F be a matrix of process Lipschitz operators and Xx the solution of (*) with initial condition x, for continuous semimartingales Za., 1 :S a :S m. Fix x, y E lRn . For r E lR there exist for every x, y E lRn with x i y (uniformly) locally bounded predictable processes Ha.(x, y), Ja.,(3(x, y), which depend on r, such that where 314 V Stochastic Differential Equations Proof. Fix x, y E IRn and let U = Xx - XY, V = F(XX)_ - F(XY)_. Ito's formula applies since U is never zero by weak injectivity (Theorem 36). Using the Einstein convention, 1lUIIT = Ilx - yilT +JrllUsII T - 2U;dU; + ~Jr{(r - 2)IJUsII T - 4u;ul +oJIlUsII T - 2 }d[Ui,Uj]s' Let (".) denote Euclidean inner product on IRn . It suffices to take (where va is the a-th column of V); and One checks that these choices work by observing that dU: = 2:::'=1 ~i,adZ't. Finally the above allows us to conclude that and the result follows. D Before giving a key corollary to Theorem 43, we need a lemma. Let jjoo be the space of continuous semimartingales X with Xo = 0 such that X has a (unique) decomposition X=N+A where N is a continuous local martingale, A is a continuous process of finite variation, No = Ao = 0, and such that [N, N]oo and Jooo IdAs I are in Loo. Further, let us define Lemma. For every p, a < 00, there exists a constant C(p, a) < 00 such that if IIXlljfoo ::; a, then 1I£(X)IIg;p ::; C(p,a). Proof Let X = N + A be the (unique) decomposition of X. Then 111£(X)II~p = E{supexp{p(Xt - -[X,X]t)}} = t 2 ::; E{ePx'} (recall that X* = sup IXt 1) t ::; E{exp{pN* +pa} } = epaE{ ePN*}, 8 Flows as Diffeomorphisms: The Continuous Case 315 since IAtl :S a, a.s. We therefore need to prove only an exponential maximal inequality for continuous martingales. By Theorem 42 of Chap. II, since N is a continuous martingale, it is a time change of a Brownian motion. That is, Nt = B[N,Nh' where B is a Brownian motion defined with a different filtration. Therefore since [N, N]oo :S a2 , we have and hence E{ePN'} :S E{exp{pB~2}}' Using the reflection principle (Theo- rem 33 of Chap. I) we have o Note that in the course of the proof of the lemma we obtained C(p, a) = 21/p exp{a + pa2 j2}. Corollary. Let -00 < r < 00 and p < 00, and let (Ar(x, Y)tk~o be as given in Theorem 43. Then £(Ar(x,y)) is locally in gP, uniformly in x,y. Proof. We need to show that there exists a sequence of stopping times T£ increasing to 00 a.s., and constants C£ < 00, such that 11£(Ar(x, y) fi II,2p :S C£ for all x,y E IRn , x =J y. - By stopping, we may assume that za and [za, Z13] are in jjOO and that IHal and IJa,13 I:S b for all (x,y), 1:S a, (3:S m. Therefore IIAr(x,y)lljjoo :S C -00 for a constant C, since if X E H and if K is bounded, predictable, then K . X E jjoo and 11K. Xlljjoo :S IIKII,2oo IIXlljjoo, as can be proved exactly analogously to Emery's inequalities (Theorem 3). The result now follows by the preceding lemma. 0 Comment. A similar result in the right continuous case is proved by a dif- ferent method in the proof of Theorem 62 in Sect. 10. Theorem 44. Let za be continuous semimartingales, 1 :S a :S m, and F an n x m matrix of process Lipschitz operators. Then the flow of the solution of is strongly injective on IRn . Proof. It suffices to show that for any compact set C C IRn , for each N, there exists an event of probability zero outside of which for every x, y E C with x =J y, inf IIX(s,w,x) - X(s,w,y)11 > O. s'5cN 316 V Stochastic Differential Equations Let x, y, s have rational coordinates. By Theorem 43 a.s. IIX(s,w,x) - X(s,w, y)/Ir = Ilx - yllr£(Ar(x, y))s. The left side of the equation is continuous (Theorem 37). As for the right side, £ (Ar (x, y)) will be continuous if we can show that the processes H';' (x, y) and J,;,(3(x, y), given in Theorem 43, verify the hypotheses of Theorem 42. To this end, let B be any relatively compact subset of IRn x IRn \ {(x,x)} (e.g., B = B l X B 2 where B l , B 2 are open balls in IRn with disjoint closures). Then Ilx - yr is bounded on B for any real number r. Without loss we take r = 1 here. Let U(x,y) = Xx -XY, V(x,y) = F(XX)_ -F(XY)_, and let V"'(x,y) be the a-th column of V. Then for (x, y) and (x', y') in B we have H"'(x,y) - H"'(x',y') (**) = (1IU(x,y)II- 2 -IIU(x',y')11- 2 )(U(x,y), V"'(x,y)) + IIU(x', y')11- 2 (U(x, y) - U(x', y'), V"'(x, y)) + IIU(x',y')11- 2 (U(x',y'), V"'(x,y) - V"'(x',y')). The first term on the right side of (**) above is dominated in absolute value by IIIU(x,y)II-IIU(x',y')III(IIU(x,y)/1 + /IU(x',y')III) IIU(x y)IIIW"'(x y)11 IIU(x, y)11 2 1IU(x', y')11 2 " ::; KIIU(x, y) - U(x', y')II(IIU(x, y)11 + IIU(x', y')II)IIU(x', y')11- 2 , where we are assuming (by stopping), that F has a Lipschitz constant K. Since U(x,y) - U(x',y') = U(x,x') - U(y,y'), the above is less than K(IIU(x, x') II + IIU(y, y')II)(IIU(x, y) II + IIU(x', y') 11)IIU(x', y') 11- 2 = K(llx - x'II£(Al(x,x')) + Ily - y'II£(Al(y,y')))· (11x - yll£(Al(x, y)) + IIx' - y'II£(AI(x', y')))llx' - y'II- 2£(A_ 2 (x', y')) ::; K111(x, y) - (x', y')II(£(Al(x, x')) + £(Al(y,y')))(£(Al(x, y)) + £(Al (x', y')))£(A_ 2 (x', y')). By the lemma following Theorem 43, and by Holder's and Minkowski's inequalities we may, for any p < 00, find stopping times T£ increasing to 00 a.s. such that the last term above is dominated in QP norm by KtII(x, y) - (x', y')11 for a constant K£ corresponding to T£. We get analogous estimates for the second and third terms on the right side of (**) by similar (indeed, slightly simpler) arguments. Therefore H'" satisfies the hypotheses of Theorem 42, for (x,y) E B. The same is true for J",,(3, and therefore Theorem 42 shows that Ar and [Ar , ArJ are continuous in (x, y) on B. (Actually we are using a local version of Theorem 42 with (x, y) E B C 1R2n instead of all of 1R2n ; this is not a problem since Theorem 42 extends to the case x E W open in IRn , because 8 Flows as Diffeomorphisms: The Continuous Case 317 Kolmogorov's Lemma does-recall that continuity is a local property.) Finally since Ar and [Ar ,ArJ are continuous in (x, y) E B we deduce that £(Ar(x, y)) is continuous in {(x,y) E jR2n: x 1= y}. We have shown that both sides of IIX(s, w, x) - X(s, w, y)llr = Ilx - yllr£(Ar(x, Y))s can be taken jointly continuous. Therefore except for a set of probability zero the equality holds for all (x, y, s) E jRn x jRn X jR+. The result follows because £(Ar(x, Y))t is defined for all t finite and it is never zero. 0 Theorem 45. Let za be continuous semimartingales, 1 ::::; a ::::; m, and let F be an n x m matrix of process Lipschitz operators. Let X be the solution of (*). Then for each N < 00 and almost all w lim inf IIX(s,w,x)11 = 00. Ilxll->oo s~N Proof By Theorem 43 the equality is valid for all r E lR. For x 1= 0 let yx = II Xx - XO 11-1. (Note that yx is well-defined by Theorem 41.) Then IYXI = Ilxll-1£(A_1(x,0)) (* * *) IYx - YYI ::::; IIXx- XYllllXx _ XOII-11IXY _ XOII-1 = Ilx - yllllxll-1Ilyll-l£(A1(x, y) )£(A_1(x, 0))£(A_1(y, 0)). Define y oo = O. The mapping x f---' xllxll- 2 inspires a distance d on jRn \ {O} by d(x, y) = IIII:I~I~IIII. Indeed, x Y 2 ( Ilx _ YII) 2 1IIIxl12- IIYl1211 = Ilxllllyll By Holder's inequality we have that 1I£(A1(x, y))£(A_ 1 (x, 0)£(A_1(y, 0)) Ilgr ::::; 11£(A1(x, y)) IIg3r 11£(A_1(x, 0)) IIg3r 11£(A_1(y, 0)) IIg3r and therefore by the corollary to Theorem 43 we can find a sequence of stop- ping times (Te)e?l increasing to 00 a.s. such that there exist constants Ce with (using (* * *)) Next set 318 V Stochastic Differential Equations yx = {yxllxll - 2 , 0 < Ilxll < 00, yeo = 0, II xii = o. Then II(Yx - yY)Tlll~r ::::; Cllix - yllr on jRn, and by Kolmogorov's Lemma (Theorem 72 of Chap. IV), there exists a jointly continuous version of (t,x) f---' ~x, on jRn. Therefore limllxll->o yx exists and equals O. Since (yx)-l = IIxxllxll-2 - XOII, we have the result. 0 Theorem 46. Let za be continuous semimartingales, 1 ::::; a ::::; m, and F be an n x m matrix of process Lipschitz operators. Let X be the solution of (*). Let cp: jRn -.jRn be the flow cp(x) = X(t,w,x). Then for almost allw one has that for all t the function cp is surjective and moreover it is a homeomorphism from jRn to jRn. Proof. As noted preceding Theorem 41, the flow cp is continuous from jRn to V n , topologized by uniform convergence on compacts; hence for a.a. w it is continuous from jRn to jRn for all t. The flow cp is injective a.s. for all t by Theorem 44. Next observe that the image of jRn under cp, denoted cp(jRn) , is closed. Indeed, let cp(jRn) denote its closure and let y E cp(jRn). Let (Xk) denote a sequence such that limk->eo CP(Xk) = y. By Theorem 45, limsuPk->eo Ilxkll < 00, and hence the sequence (Xk) has a limit point x E jRn. Continuity implies cp(x) = y, and we conclude that cp(jRn) = cp(jRn); that is, cp(jRn) is closed. Then, as we have seen, the set {xd is bounded. If Xk does not converge to x, there must exist a limit point z =I x. But then cp(z) = y = cp(x), and this violates the injectivity, already established. Therefore cp-l is continuous. Since cp is a homeomorphism from jRn to cp(jRn) , the subspace cp(jRn) of jRn is homeomorphic to a manifold of dimension n in jRn; therefore by the theorem of the invariance of the domain (see, e.g., Greenberg [84, page 82]), the space cp(jRn) is open in jRn. But cp(jRn) is also closed and non-empty. There is only one such set in jRn that is open and closed and non-empty and it is the entire space jRn. We conclude that cp(jRn) = jRn. 0 Comment. The proof of Theorem 46 can be simplified as follows: extend cp to the Alexandrov compactification jR~ = jRn U {oo} of jRn to rp by "ip(x) = {cp(X), 00, x E jRn, x = 00. Then rp is continuous on jR~ by Theorem 45, and obviously it is still injective. Since jR~ is compact, rp is a homeomorphism of jR~ onto rp(jR~). However jR~ is topologically the sphere sn, and thus it is not homeomorphic to any proper subset (this is a consequence of the Jordan-Brouwer Separation Theorem (e.g., Greenberg [84, page 79]). Hence "ip(jR~) = jR~. 8 Flows as Diffeomorphisms: The Continuous Case 319 We next turn our attention to determining when the flow is a diffeomor- phism of ]R.n. Recall that a diffeomorphism of]R.n is a bijection (one to one and onto) which is Coo and which has an inverse that is also Coo. Clearly the hypotheses on the coefficients need to be the intersection of those of Sect. 7 and process Lipschitz. First we introduce a useful concept, that of right stochastic exponentials, which arises naturally in this context. For given n, let Z be an n x n matrix of given semimartingales. If X is a solution of where X is an n x n matrix of semimartingales and I is the identity matrix, then X = £(Z), the (matrix-valued) exponential of Z. Since the space of n x n matrices is not commutative, it is also possible to consider right stochastic integrals, denoted where Z is an n x n matrix of semimartingales and H is an n x n matrix of (integrable) predictable processes. If I denotes matrix transpose, then (Z : H) = (H' . Z')/, and therefore right stochastic integrals can be defined in terms of stochastic integrals. Elementary results concerning right stochastic integrals are collected in the next theorem. Note that J Y_dZ and [Y, ZJ denote n x n matrix-valued processes here. Theorem 47. Let Y, Z be given n x n matrices of semimartingales, H an n x n matrix of locally bounded predictable processes. Then, (i) ytZt - YoZo = J~ Ys_dZs + J~(dYs)Zs- + [Y, Z]t; (ii) [H· Y, Z] = H· [Y, Z]; and (iii) [Y, Z: HJ = [Y, ZJ : H. Moreover if F is an n x n matrix of functional Lipschitz operators, then there exists a unique n x n matrix of ]]J)-valued processes which is the solution of Proof The first three identities are easily proved by calculating the entries of the matrices and using the results of Chap. II. Similarly the existence and uniqueness result for the stochastic integral equation is a simple consequence of Theorem 7. 0 Theorem 47 allows the definition of the right stochastic exponential. 320 V Stochastic Differential Equations Definition. The right stochastic exponential of an n x n matrix of semi- martingales Z, denoted £R(Z), is the (unique) matrix-valued solution of the equation X t = I + it (dZs)Xs_. We illustrate the relation between left and right stochastic exponentials in the continuous case. The general case is considered in Sect. 10 (see Theo- rem 63). Note that £R(Z) = £(Z')'. Theorem 48. Let Z be an n x n matrix of continuous semimartingales with Zo = O. Then £(Z) and £R(-Z + [Z, Z]) are inverses; that is, £(Z)£R(-Z + [Z, Z]) = I. Proof Let U = £(Z) and V = £R(_Z + [Z,Z]). Since UoVo = I, it suffices to show that d(Ut Vi) = 0, all t > O. Note that dV = (-dZ + d[Z, Z])V, (dU)V = (UdZ)V, and d[U, Vj = Ud[Z, V] = -Ud[Z,ZJV. Using Theorem 47 and the preceding, d(UV) = UdV + (dU)V + d[U, V] = U( -dZ + d[Z, Z])V + UdZV - Ud[Z, ZJV =0, and we are done. o The next theorem is a special case of Theorem 40 (of Sect. 7), but we state it here as a separate theorem for ease of reference. Theorem 49. Let (Zl, ... , zm) be continuous semimartingales with zg = 0, 1 ::; i ::; m, and let U~), 1 ::; i ::; n, 1 ::; a ::; m, be functions mapping lRn to lR, with locally Lipschitz partial derivatives up to order N, 1 ::; N ::; 00, and bounded first derivatives. Then there exists a solution X (t, w, x) to such that its flow 'P : x -> X(x, t,w) is N times continuously differentiable on lRn . Moreover the first partial derivatives satisfy the linear equation (1::; i ::; n). where 61. is Kronecker's delta. 9 General Stochastic Exponentials and Linear Equations 321 Observe that since the first partial derivatives are bounded, the coefficients are globally Lipschitz and it is not necessary to introduce an explosion time. Also, the value N = 00 is included in the statement. The explicit equation for the partial derivatives comes from Theorem 39. Let D denote the n x n matrix-valued process (0) The process D is the right stochastic exponential ER(y), where Y is defined by dyi,j = ~ af~ (X )dZa s L..J ax. S S • a=l J Combining Theorems 48 and 49 and the above observation we have the im- portant following result. Theorelll 50. With the hypotheses and notation of Theorem 49, the matrix Dt is non-singular for all t > 0 and x E jRn, a.s. Theorelll 51. Let (Zl, ... , zm) be continuous semimartingales and let (J~), 1 ::::; i ::::; n, 1 ::::; a ::::; m, be functions mapping jRn to jR, with partial derivatives of all orders, and bounded first partials. Then the flow of the solution of x: = Xi + f it f~(Xs)dZ':, 1::::; i::::; n, a=l 0 is a diffeomorphism from jRn to jRn. Proof. Let cp denote the flow of X. Since (J~h~i~n,l~a~m have bounded first partials, they are globally Lipschitz, and hence there are no finite explosions. Moreover since they are Coo, the flow is Coo on jRn by Theorem 49. The coefficients (f~) are trivially process Lipschitz, hence by Theorem 46 the flow cp is a homeomorphism; in particular it is a bijection of jRn. Finally, the matrix Dt (defined in (0) preceding Theorem 50) is non-singular by Theorem 50, thus cp-1 is Coo by the Inverse Function Theorem. Since cp-1 is also Coo, we conclude cp is a diffeomorphism of jRn. 0 9 General Stochastic Exponentials and Linear Equations Let Z be a given continuous semimartingale with Zo = 0 and let E(Z)t denote the unique solution of the stochastic exponential equation X t = 1 + it XsdZs. Then X t = E(Z)t = exp{Zt - ~[Z, Zld (ef., Theorem 37 of Chap. II). It is of course unusual to have a closed form solution of a stochastic differential 322 V Stochastic Differential Equations equation, and it is therefore especially nice to be able to give an explicit solution of the stochastic exponential equation when it also has an exogenous driving term. That is, we want to consider equations of the form , where HE ]]J) (cadlag and adapted), and Z is a continuous semimartingale. A unique solution of (**) exists by Theorem 7. It is written EH(Z). Theorem 52. Let H be a semimartingale and let Z be a continuous semi- martingale with Zo = O. Then the solution EH(Z) of equation (**) is given by EH(Z)t = E(Z)dHo + r t E(Z);ld(Hs - [H, Z]s)}.Jo+ Proof We use the method of "variation of constants." Assume the solution is of the form X t = GtUt , where Ut = E(Z)t, the normal stochastic exponential. The process G is cadlag while U is continuous. Using integration by parts, dXt = Gt_dUt + UtdGt + d[G, U]t = Gt- UtdZt + UtdGt + Utd[G, Z]t = Xt_dZt + Utd{Gt + [G,Z]t}. If X is the solution of (**), then equating the above with (**) yields or dHt = Utd{Gt + [G, Z]t}. Since U is an exponential it is never zero and l/U is locally bounded. Therefore 1 -dHt = dGt + d[G, ZJt.Ut Calculating the quadratic covariation of each side with Z and noting that [[G, Z], Z] = 0, we conclude 1[U ·H,Z] = [G,Z]. Therefore equation (* * *) becomes 1 1 UdH = dG + Ud[H, Z], and Gt = J~ Us-1d(Hs - [H, Z]s). Recall that Ut = E(Z)t and X t = GtUt , and the theorem is proved. 0 9 General Stochastic Exponentials and Linear Equations 323 Since E(Z)t 1 = 1jE(Z)t appears in the formula for EH(Z), it is worthwhile to note that (for Z a continuous semimartingale) d (_1_) = dZ - d[Z, ZJ E(Z) E(Z) and also 1 E(Z) = E(-Z + [Z, Z]). A more complicated formula for EH(Z) exists when Z is not continuous (see Yoeurp-Yor [236]). The next theorem generalizes Theorem 52 to the case where H is not necessarily a semimartingale. Theorem 53. Let H be cadlag, adapted (i. e., H E ]]J»), and let Z be a con- tinuous semimartingale with Zo = o. Let X t = EH(Z)t be the solution of X t = Ht +I t Xs_dZs. Then X t = EH(Z)t is given by Proof Let yt = X t - Ht . Then Y satisfies yt = it Hs_dZs + it Ys_dZs = K t +I t Ys_dZs, where K is the semimartingale H_ . Z. By Theorem 52, yt = E(Z)dKo +it E(Z)-;ld(Ks - [K, ZJs)} 0+ and since K o = 0 and [K, Z]t = J~ Hs_d[Z, Z]s, from which the result follows. o Theorem 54 uses the formula of Theorem 52 to give a pretty result on the comparison of solutions of stochastic differential equations. 324 V Stochastic Differential Equations Lelllllla. Suppose that F is functional Lipschitz such that if Xt(w) = 0, then F(X)t-(w) > 0 for continuous processes X. Let C be a continuous increasing process and let X be the solution of with Xo > O. Then prj t > 0 : X t ::::; O} = O. Proof. Let T = inf{t > 0 : X t = O}. Since X o :::: 0 and X is continuous, X s :::: 0 for all s < Ton {T < oo}. The hypotheses then imply that F(X)r- > 0 on {T < oo}, which is a contradiction. 0 COllllllent. In the previous lemma if one allows Xo = 0, then it is necessary to add the hypothesis that C be strictly increasing at O. One then obtains the same conclusion. Theorelll 54 (Colllparison Theorelll). Let (zah::::a::::m be continuous semimartingales with zg = 0, and let Fa be process Lipschitz. Let A be a continuous, adapted process with increasing paths, strictly increasing at t = O. Let G and H be process Lipschitz functionals such that G(X)t_ > H(X)t_ for any continuous semimartingale X. Finally, let X and Y be the unique solutions of X t = Xo + r t G(X)s_dAs + r t F(X)s_dZs,Jo+ Jo ft = Yo +it H(Y)s_dAs +it F(Y)s_dZs 0+ 0 where Xo :::: Yo and F and Z are written in vector notation. Then prj t > 0 : X t ::::; Y;;} = O. Proof. Let Ut = X t - ft, Nt = I t{F(X)s_ - F(Y)s-}(Xs - Ys)-ll{xs ;"Ys}dZs, and Ct = Xo - Yo + r t {G(X)s_ - H(Y)s-}dAs.Jo+ Then Ut = Ct + J~ Us_dNs, and by Theorem 52 Next set 9 General Stochastic Exponentials and Linear Equations 325 1 Vi = £(N)t Ut , and define the operator K on continuous processes W by K(W) = G(W£(N) + Y) - H(Y). Note that since G and H are process Lipschitz, if W t = 0 then G(W£(N) + Y)t- = G(Y)t-. Therefore K has the property that W t (w) = 0 implies that K(W)t- > O. Note further that K(V) = G(U + Y) - H(Y) = G(X) - H(Y). Next observe that 1 Vi = £(N)t Ut = Xo - Yo + t £(N);ldCsJo+ = Xo - Yo + r t £(N);l{G(X)s_ - H(Y)s-}dAsJo+ = Xo - Yo + t K(V)s_£(N);ldAsJo+ t = Xo - Yo +1K(V)s_dDs, where Dt = J~+ £(N);ldAs. Then D is a continuous, adapted, increasing process which is strictly increasing at zero. Since V satisfies an equation of the type given in the lemma, we conclude that a.s. Vi > 0 for all t. Since £(N);:l is strictly positive (and finite) for all t ~ 0, we conclude Ut > 0 for all t > 0, hence X t > yt for all t > O. 0 COInment. If Xo > Yo (Le., Xo = Yo is not allowed), then the hypothesis that A is strictly increasing at 0 can be dropped. The theory of flows can be used to generalize the formula of Theorem 52. In particular, the homeomorphism property is used to prove Theorem 55. Consider the system of linear equations given by where H = (Hi), 1 :::; i :::; n is a vector of n semimartingales, X takes values in lR.n , and Aj is an n x n matrix of adapted, cadlag processes. The processes Zj, 1 :::; j :::; m, are given, continuous semimartingales which are zero at zero. Define the operators Fj on lIJ)n by F(X)t = Aixt where Aj is the n x n matrix specified above. The operators Fj are essentially process Lipschitz. (The Lipschitz processes can be taken to be IIAi II which 326 V Stochastic Differential Equations is cadla,g, not caglad, but this is unimportant since one takes F(X)t_ in the equation.) Before examining equation (*4), consider the simpler system (1 ::::: i, k ::::: n), Ui,k = 8i +~ it ~(Aj) U€,kdZj t k L L >,€ 8- 8- 8 j=l 0 €=1 where 8~ = {I, 0, i = k, i # k. Letting I denote the n x n identity matrix and writing the preceding in matrix notation yields Ut = 1+ f it A~_U8_dzt, j=l 0 where U takes its values in the space of n x n matrices of adapted processes in ]]J). Theorem 55. Let Aj, 1 ::::: j ::::: m, be n x n matrices of cadlag, adapted processes, and let U be the solution of (*5). Let X[ be the solution of (*4) where Ht = x, x E jRn. Then X[ = Utx and for almost all w, for all t and x the matrix Ut (w) is invertible. Proof Note that Ut is an n x n matrix for each (t,w) and x E jRn, so that Utx is in jRn. If Xt = Utx, then since the coefficients are process Lipschitz we can apply Theorem 46 (which says that the flow is a homeomorphism of jRn) to obtain the invertibility of Ut(w). Note that U is also a right stochastic exponential. Indeed, U = £R(V), where Vi = J~ 'L';=1 A~_dZL and therefore the invertibility also follows from Theorem 48. Thus we need to show only that Xt = Utx. Since Utx solves (*4) with Ht = x, we have Utx = Xt a.s. for each x. Note that a.s. the function x f---' U(w)x is continuous from jRn into the subspace of V n consisting of continuous functions; in particular (t, x) f---' Ut (w)x is continuous. Also as shown in the proof of Theorem 46, (x, t) f---' Xt is continuous in x and right continuous in t. Since Utx = Xt a.s. for each fixed x and t, the continuity permits the removal of the dependence of the exceptional set on x and t. 0 Let U- 1 denote the n x n matrix-valued process with continuous trajec- tories a.s. defined by (U-1 Mw) = (Ut (w)) -1 . Recall equation (*4) 9 General Stochastic Exponentials and Linear Equations 327 where H is a column vector of n semimartingales and zg = 0. Let [H, Zj] denote the column vector of n components, the ith one of which is [Hi, Zj]. Theorem 56. Let H be a column vector of n semimartingales, zj (1 ::; j ::; m) be continuous semimartingales with zg = 0, and let Aj, 1 ::; j ::; m be n x n matrices of processes in j[]). Let U be the solution of equation (*5). Then the solution x H of (*4) is given by Proof Write X H as the matrix product UY. Recall that U- 1 exists by Theo- rem 48, hence Y = U- 1X H is a semimartingale, that we need to find explicitly. Using matrix notation throughout, we have m d(UY) = dH + L A~X_dZj. j=l Integration by parts yields (recall that U is continuous) m (dU)Y- + U(dY) + d[U, Yj = dH + LA~U_Y_dZj, j=l by replacing X with UY on the right side above. However U satisfies (*5) and therefore m (dU)Y- = LA~U_Y_dZj, j=l and combining this with the preceding gives U(dY) + d[U, Y] = dH, or equivalently dY = U-1dH - U-1d[U, Y]. Taking the quadratic covariation of the preceding equation with Z, we have dry, zjj = U-1d[H, zj], since [U- 1d[U, Y], Zj] = 0, 1 ::; j :::; m. However since U satisfies (*5), m m d[U,Y] = LA~U_d[Y,Zj] = LA~UU-1d[H,Zj] j=l j=l m = LA~d[H,Zj], j=l 328 V Stochastic Differential Equations since U equals U_. Substitute the above expression for d[U, Y] into (*6) and we obtain m dY = U-1(dH - LA~d[H,Zj]), j=l and since X H = UY, the theorem is proved. 10 Flows as Diffeomorphisms: The General Case In this section we study the same equations as in Sect. 8, namely o except that the semimartingales (Zh 10 Flows as Diffeomorphisms: The General Case 329 may also be written where X t and x are column vectors in IRn , f(Xs-) is an n x m matrix, and Z is a column vector of m semimartingales. Hypothesis (H3). f is Coo and has bounded derivatives of all orders. Note that by Theorem 40 of Sect. 7, under (H3) the flow is Coo. The key to studying the injectivity (and diffeomorphism properties) is an analysis of the jumps of the semimartingale driving terms. Choose an E > 0, the actual size of which is yet to be determined. For (Z"'h 330 V Stochastic Differential Equations Proof By Theorem 57, the solution (*) can be constructed by composition of functions of the types given in the theorem. Since the composition of diffeo- morphisms is a diffeomorphism, the theorem is proved. 0 We begin by studying the functions x --. X? (x, w) and x --. X~+l(X, w). J Note that by our construction and choice of the times Tj , we need only to consider the case where Z = zj has a norm in H oo smaller than c. The following classical result, due to Hadamard, underlies our analysis. Theorem 59 (Hadamard's Theorem). Let 9 : IRn --.lRn be Coo. Suppose (i) limllxll-+oo Ilg(x)11 = 00, and (ii) the Jacobian matrix g'(x) is an isomorphism oflRn for all x. Then 9 is a diffeomorphism of IRn . Proof By the Inverse Function Theorem the function 9 is a local diffeomor- phism, and hence it suffices to show it is a bijection of IRn . To show that 9 is onto (i.e., a surjection), first note that g(lRn ) is open and non-empty. It thus suffices to show that g(lRn ) is a closed subset of IRn , since IRn itself is the only nonempty subset of IRn that is open and closed. Let (Xik:~1 be a sequence of points in IRn such that limi-+oo g(Xi) = Y E IRn . We will show that y E g(lRn ). Let Xi = tiVi, where ti > 0 and Ilvi II = 1. By choosing a subsequence if necessary we may assume that Vi converges to V E sn, the unit sphere, as i tends to 00. Next observe that the sequence (t i )i>l must be bounded by condition (i) in the theorem: for if not, then t i = Ilxi II tends to 00 along a subsequence and then Ilg(Xik)11 tends to 00 by (i), which contradicts that limi-+oo g(Xi) = y. Since (t i )i21 is bounded we may assume limi-+oo ti = to E IRn again by taking a subsequence if necessary. Then limi-+oo Xi = tov, and by the continuity of 9 we have y = limi-+oo g(Xi) = g(tov). To show 9 is injective (i.e., one-to-one), we first note that 9 is a local homeomorphism, and moreover 9 is finite-to-one. Indeed, if there exists an infinite sequence (Xn )n21 such that g(xn ) = Yo, all n, for some Yo, then by condition (i) the sequence must be bounded in norm and therefore have a cluster point. By taking a subsequence if necessary we can assume that X n tends to x (the cluster point), where g(xn ) = Yo, all n. By the continuity of 9 we have g(x) = Yo as well. This then violates the condition that 9 is a local homeomorphism, and we conclude that 9 is finite-to-one. Since 9 is a finite-to-one surjective homeomorphism, it is a covering map.16 However since IRn is simply connected the only covering space of IRn is IRn (the fundamental group of IRn is trivial). Therefore the fibers g-l(x) for x E IRn each consist of one point, and 9 is injective. 0 The next step is to show that the functions x 1---4 X? (x, w) and x 1---4 J X~+1(X, w) of Theorem 58 satisfy the two conditions of Theorem 59 and are 16 For the algebraic topology used here, the reader can consult, for example, Munkries [183, Chapter 8]. 10 Flows as Diffeomorphisms: The General Case 331 thus diffeomorphisms. This is done in Theorems 62 and 64. First we give a result on weak injeetivity which is closely related to Theorem 41. Theorem 60. Let za be semimartingales, 1 :::; a:::; m with Zo = 0, and let F be an n x m matrix of process Lipschitz operators with non-random Lipschitz constant K. Let Hi E JDl, 1:::; i:::; n (cadlag, adapted). IfL.:=lllzalllioo < E, for E > 0 sufficiently small, then the flow of the solution of - is weakly injective. 17 Proof Let x, y E IRn , and let Xx, XY denote the solutions of the above equa- tion with initial conditions x, y, respectively. Let u = x - y, U = Xx - XY, and V = F(XX)_ - F(XY)_. Then VEL and IVI :::; KIU_I. Also, Therefore t::..Us = L.a Vsa t::..ZC: and moreover (using the Einstein convention to leave the summations implicit) II!::..Usll ::::: II~allllt::..ZC:II ::::: CIIUs_IIE 1 < 2 11Us- 1I if E is small enough. Consequently IIUs II ;? ~llUs-ll. Define T = inf{t > 0 : Ut - = O}. Then Ut - =1= 0 on [0, T) and the above implies Ut =1= 0 on [0, T) as well. Using Ito's formula for f(x) = log Ilxll, as in the proof of Theorem 41 we have For s fixed, let 17 We are using vector and matrix notation, and the Einstein convention on sums. The Einstein convention is used throughout this section. 332 V Stochastic Differential Equations so that the last sum on the right side of equation (**) can be written Lo °sufficiently small, then for r E IR there exist uniformly TOcally bounded predictable processes Ha(x, y) and Ka,f3(x, y), which depend on r, such that /lXx - XY/I" = Ilx - YII"[(A,.(x, y)) where XX is the solution of The semimartingale A,. is given by 10 Flows as Diffeomorphisms: The General Case 333 where Jt = Lo 334 V Stochastic Differential Equations K~,/1(x, y) = ~r{(r - 2)IIUs_11 4 (Us_, Vsa)(Us_, V!) + IIUs_II- 2 (Vsa, V!)}l{Us_#O} , as in the proof of Theorem 43. Note that 1{Us_#o} is indistinguishable from the zero process by weak injectivity (Theorem 60). These choices for H a and Ka,/1 are easily seen to work by observing that U; = 2::;;=1 J; ~i,adZ':, and the preceding allows us to conclude that and the result follows. o Theorem 62. Let (Zah::;a::;m be semimartingales, Zff = 0, F an n x m matrix of process Lipschitz operators with a non-random Lipschitz constant, and Hi E [J), 1::; i ::; n. Let X = X(t,w, x) be the solution of Xt= x + it F(X)s_dZs. If 2::;:1 Ilzalllioo < E for E > 0 sufficiently small, then for each N < 00 and almost all W - lim inf IIX(s,w,x)11 = 00.IIxlI-+eo s::;N Proof. The proof is essentially the same as that of Theorem 45, so we only sketch it. For x =1= 0 let yx = IIXx - XO 11- 1 , which is well-defined by weak injectivity (Theorem 60). Then IY X - YYI ::; IIX x - XYIIIIXX - XOII- 1 1IXY _ XOIl- 1 = Ilx - yllllxll- 1I1yll-1E(A I (x, y) )E(A_ 1(x, 0))E(A_ 1(y, 0)) by Theorem 61, where A,,(x, y) is as defined in Theorem 61. Set yeo = O. Since IIZllHoo < E each za has jumps bounded by E, and the process Jt defined in Th-;orem 61 also has jumps bounded by C"E2 . Therefore we can stop the processes A" (x, y) at an appropriately chosen sequence of stopping times (Tek,,:l increasing to 00 a.s. such that each A,,(x, y) E S(E) for a given E, and for each £, uniformly in (x, y). However if Z is a semimartingale in S(E), then since E(A,,(x, y)) satisfies the equation Ut = 1 + J;+ Us_dA,,(x, Y)s, by Lemma 2 of Sect. 3 of this chapter we have IIE(A,,(x, y))lIsp ::; C(p, z) < 00, 10 Flows as Diffeomorphisms: The General Case 335 where C(p, z) is a constant depending on p and z = IIA,.(x, y)lllIoo ::::: ke, the bound for £, provided of course that c is sufficiently small. We conclude that for these Te there exist constants Ce such that where p > n, and where d is the distance on IRn \ {o} given by d(x, y) = il':'~'~"'I' Set yx = {YXIIXII- 2 , 0 < Ilxll < 00, yeo = 0, Ilxll = o. Then II(Yx - YYflll~p ::::: Cfllx - yilP on IRn , and by Kolmogorov's Lemma (Theorem 72 of Chap. IV) we conclude that limllxll-+o yx exists and it is zero. Since (yX)-l = IIxxllxlI-2 - XOII, the result follows. 0 If cp is the flow of the solution of (*), Theorem 62 shows that lim IIcp(x)11 = +00, Ilxll-+eo and the first condition in Hadamard's Theorem (Theorem 59) is satisfied. Theorem 63 allows us to determine when the second condition in Hadamard's Theorem is also satisfied (see Theorem 64), but it has an independent interest. First, however, some preliminaries are needed. For given n, let Z be an n x n matrix of given semimartingales. Recall that X = [(Z) denotes the (matrix-valued) exponential of Z, and that [R(Z) de- notes the (matrix-valued) right stochastic exponential of Z, which was defined in Sect. 8, following Theorem 47. Recall that in Theorem 48 we showed that if Z is an n x n matrix of continuous semimartingales with Zo = 0, then [(Z)[R( -Z + [Z, Z]) = I, or equivalently [(-Z + [Z, Z])[R(Z) = I. The general case is more delicate. Theorem 63. Let Z be an n x n matrix of semimartingales with Zo = o. Suppose that Wt = -Zt + [Z, Z]f + LO 336 V Stochastic Differential Equations (dU)V_ = U_( -dZ + d[Z, Z]C + dJ)V_, U_dV = U_(dZ)V_, d[U, V] = U_d[W; zlV- = -U_d[Z, zlV- + U_d[J, zlV-, since d[Z, [Z, Z]C] = O. By Theorem 47 d(UV) = U_dV + (dU)V_ + d[U, V]; using the above calculations several terms cancel, yielding [J, Z]t = L b.Jsb.Zs = L (b.Zs)3(I + b.Zs)-\ O 0 and therefore the process Wof Theorem 63 is a well-defined semimartingale. 0 Theorem 64. Let (Zah 10 Flows as Diffeomorphisms: The General Case 337 Proof By Theorem 39 (in Sect. 7) the Jacobian matrix D satisfies the right stochastic exponential equation i t afiDt = I + (!::> Q(Xs- )dZ~)Ds_,o UXk and the matrix semimartingale differential ~ (Xs - )dZ': satisfies the hy- potheses of the corollary of Theorem 63, whence the result. D Before stating the principal result of this section, we need to define two subsets of lRm ; recall that under Hypotheses (HI), (H2), and (H3), that Z = (ZQh~Q~m is a given m-tuple of semimartingales and that f(x) = (f~(x» is an n x m matrix of Coo functions. Let v = {z E lRm : H(x) = x + f(x)z is a diffeomorphism of lRn } I = {z E lRm : H(x) = x + f(x)z is injective in lRn }. Clearly V C I. Theorem 65. Let Z and f be as given in Hypotheses (Hi), (H2), and (H3), and let X be the solution of X t = X + it f(Xs-)dZs. The flow of X is a.s. a diffeomorphism of lRn (resp. trajectories of X from different initial points a.s. never meet) for all t if and only if all the jumps of Z belong to V (resp. all the jumps of Z belong to I). Proof Recall the processes X? (x,w) and X;,+l(X,w) defined in Theorem 57, J and the linkage operator Hj (x) = x + f(x)LlZT;, define~ immediately preced- ing Theorem 57. By hypothesis the linkage operators HJ(x) are clearly diffeo- morphisms of lRn (resp. injective), and by Theorems 62 and 64, Hadamard's conditions are satisfied (Theorem 59), and therefore the functions x f---> X? (x,w) and x f---> X;,+1(x,w) are diffeomorphisms of lRn if c > 0 is taken J small enough in the definition of the stopping times (Tj k:?l, which it is always possible to do. Therefore by Theorem 58 the flow 'P : x -+ Xt(x,w) is a.s. a diffeomorphism of lRn for each t > O. The necessity is perhaps the more surprising part of the theorem. First observe that by Hadamard's Theorem (Theorem 59) the set V contains a neighborhood of the origin. Indeed, if z is small enough and x is large enough then Ilf(x)zll ::; Ilxll/2 since f is Lipschitz, which implies that IIH(x)11 2: Ilxll/2 and thus condition (i) of Hadamard's Theorem is satisfied. On the other hand H'(x) = 1+ j'(x)Z is invertible for all x for Ilzll small enough because f'(x) is bounded; therefore condition (ii) of Hadamard's Theorem is satisfied. Since f(x) is Coo (Hypothesis (H3», we conclude that V contains a neighborhood of the origin. 338 V Stochastic Differential Equations To prove necessity, set r 1 = {W : :3 S >°with ~Zs(w) E VC}, r2 = {w : :3 s >°with ~Zs(w) E I C }. Suppose p(r1 ) > 0. Since V contains a neighborhood of the origin, there exists an c >°such that all the jumps of Z less than c are in V. We also take c so small that all the functions X f---> Xf i (x) are diffeomorphisms as soon as the linkage operators H k are, all k ~ i. Since the jumps of Z smaller than c are in V, the jumps of Z that are in VC must take place at the times Ti . Let Aj = {w : ~ZTi E V, all i < i, and ~ZTj E VC}. Since p(r1 ) > 0, there must exist a i such that P(Aj) > 0. Then for wE Aj, X f---> XTj_(X,W) is a diffeomorphism, but Hj(x,w) is not a diffeomorphism. Let Wo E Aj and to = Tj(wo). Then X f---> Xto(x,wo) is not a diffeomorphism, and therefore P{w : :3 t such that x ----> Xt(x,w) is not a diffeomorphism} > 0, and we are done. The proof of the necessity of the jumps belonging to I to have injectivity is analogous. D Corollary. Let Z and f be as given in Hypotheses (HI), (H2), and (H3), and let X be the solution of X t = X +it f(Xs-)dZs. Then different trajectories of X can meet only at the jumps of Z. Proof We saw in the proof of Theorem 65 that two trajectories can intersect only at the times Tj that slice the semimartingales ZQ into pieces of HOG norm less than c. If the ZQ do not jump at T jo for some io, however, and paths of X intersect there, one need only slightly alter the construction of Tjo (d., the proof of Theorem 5, where the times T j were constructed), so that Tjo is not included in another sequence that c-slices (ZQh:S;Q:S;m, to achieve a contradiction. (Note that if, however, (ZQh:S;Q:S;m has a large jump at Tjo , then it cannot be altered.) D 11 Eclectic Useful Results on Stochastic Differential Equations We begin this collection of mostly technical results concerning stochastic dif- ferential equations with some useful moment estimates. And we begin the moment estimates with preliminary estimates for stochastic integrals. The first result is trivial, and the second is almost as simple. 11 Eclectic Useful Results on Stochastic Differential Equations 339 Lemma. For any predictable (matrix-valued) process H and for any p > 1, and for 0 :::; t :::; 1, we have Proof Since 0 :::; t :::; 1, we have that ds on [0,1] is a probability measure, and since f(x) = x P is convex for p > 1, we can use Jensen's inequality. The result then follows from Fubini's Theorem. 0 Lemma. For any predictable (matrix-valued) process H and multidimen- sional Brownian motion B, and for any p 2': 2, and for 0 :::; t :::; 1, there exists a constant cp depending only on p such that Proof The proof for the case p = 2 is simply Doob's inequality; we give it for the one dimensional case. where the last equality is by Fubini's Theorem. For the case p > 2, we use Burkholder's inequality (see Theorem 48 of Chap. IV, where the last inequality follows from Jensen's inequality (recall that p 2': 2 so that f(x) = x p / 2 is a convex function, and since 0 :::; t :::; 1 we have used that ds on [0, 1] is a probability measure) and Fubini's Theorem. 0 Theorem 66. Let Z be a d-dimensional Levy process with E{II Zt IILP} < 00 for 0 :::; t :::; 1, H a predictable (matrix-valued) process and p 2': 2. Then there exists a finite constant K p such that for 0 :::; t :::; 1 we have Proof We give the proof for the one dimensional case. I8 Since the Levy pro- cess Z can be decomposed Zt = bt + (JBt + Mt , where M is a purely dis- continuous martingale (that is, M is orthogonal in the L 2 sense to the stable subspace generated by continuous L 2 martingales), it is enough to prove the inequality separately for Zt = bt, and Zt = (JBt> and Zt = M t. The preceding two lemmas prove the first two cases, so we give the proof of only the third. 18 An alternative (and simpler) proof using random measures can be found in [105]. 340 V Stochastic Differential Equations In the computations below, the constants Cp and the functions K p (') vary from line to line. Choose the rational number k such that 2k ~ P < 2k +1. Applying the Burkholder-Gundy inequalities19 for p 22 we have Set aM := E{[M, Mh} = E{2: I~MsI2} = JIxI 2vM(dx) < 00. s:::::l Since [M, M] is also a Levy process, we have that [M, M]t - aMt is also a martingale. Therefore the inequality above becomes We apply a Burkholder-Gundy inequality again to the first term on the right side above to obtain E{ll t HsdMsIP} ~CpE{(2: IHs~MsI4)P/4} o s:::::t +Kp(t)[J !xI 2vM(dx)]P/2E{l t IHsIPds}. We continue recursively to get E{ll t HsdMsjP} ~CpE{(2: IHs~MsI2k+l)p/2k+l} o s:::::t k t + Kp(t)(~[J Ixl 2i vM(dX)]P2-i)E{1IHsIPds}. Next we use the fact that, for any sequence a such that IlallIq is finite, Ilallz2 ~ IlallIq for 1 ~ q ~ 2. As 1 :s: p2- k < 2 we get [2: IHs~MsI2k+l]P/2k+l = [2:(IHs~MsI2k)2]h1< s:::::t s:::::t whence 19 The Davis inequality is for the (important) case p = 1 only. 11 Eclectic Useful Results on Stochastic Differential Equations 341 E{{:L IHs~MsI2k+l}p/2k+l} ::; E{:L IHs~MsIP}. s~t s~t Note that l:s 342 V Stochastic Differential Equations Theorem 66 can be used to prove many different results concerning solu- tions of stochastic differential equations driven by Levy processes. We give two examples. (These types of results are true more extensively (for example in n dimensions) with similar proofs; see for example [105].) Our first example is a moment estimate. As a convenience, we continue to work on the time interval [0,1]. Theorem 67. Let (J be continuously differentiable with a bounded derivative, and let Z be a Levy process with Levy measure v such that ~XI>1 IxIPv(dx) < 00, which is equivalent to E{IZtIP} < 00 for p ?': 2 and for all t, 0 ~ t ~ 1. Let XX denote the unique solution of the stochastic differential equation Then we have E{sup IX;IP } ~ K(1 + IxIP) 8 where K is of course a positive constant. Before proving this theorem we need to recall Gronwall's inequality.20 Theorem 68 (Gronwall's Inequality). Let a be a function from IR+ to itself, and suppose a(s) ~ c + k18 a(r)dr < 00 for 0 ~ s ~ t. Then a(t) ~ cekt . Moreover if c = 0 then a vanishes identically. The proof is simple and usually taught in elementary differential equations courses. Proof of Theorem 67. This is a consequence of Theorem 66 and Gronwall's inequality. 0 The next corollary follows easily from Theorem 67. Theorem 69. Let (J and Z be as in Theorem 67. Let 9 be continuous and for q > 0 define II 9 Ilq= inf{a > 0 : Ig(x)1 ~ a(1 + Ixl q )}· For q = 0we let II gilD be the LOO norm. If Pt denotes the transition semigroup of the Markov process Xx of equation (*), then we have for q ?': 0 20 Extensions of this classic version of Gronwall's inequality are in the exercises for this chapter. See also [182]. 11 Eclectic Useful Results on Stochastic Differential Equations 343 We now turn to our second example, which consists of moment estimates on the flows of the solution Xx of equation (*). We first establish some nota- tion. For a Levy process Z with Levy measure 1/ and with finite p-th moment for p 2 2, we decompose it as Zt = bt +eBt +M t where B is a standard Brow- nian motion and M is a martingale independent from B and L 2 orthogonal to all continuous martingales. We then define which is finite since by assumption E{IZtIP} < 00 and p 22. Here we consider the multidimensional case: both Z and the solution Xx of equation (*) are assumed to be d-dimensional, where d 2 1.21 We let X:,(k) represent a k-th derivative of the flow of Xx. Theorem 70. For k 2 1 assume that (1' is differentiable k times and that all partial derivatives of (1' of order greater than or equal to one are bounded. Assume also that Z has moments of order kp with p 2 2. With the nota- tion defined right before this theorem, we have that there exists a constant K(k,p, (1', 'TIp) such that for 0::::; s ::::; 1, E{suPIX:,(k)IP}::::; K. s Proof We recall from Theorem 39 that Xx together with xx,(k) constitute the unique solution of the system of equations Xi = Xi +f it (1'~(Xs_)dZ~ Q=1 0 m n t i Dtt = 8t + L L 1~:Q (Xs-)Dks_ dZ':, Q=1 j=1 0 J (D) as long as k 2 2, and with a slightly simpler formulation if k = 1. (Note that this is the same equation (D) as on page 305.) We can write the second equation above in a slightly more abstract way if k 2 2; and if k = 1 the equation is the simpler 21 One could also allow the dimensions of Z and Xx to be different, but we do not address that case here. 344 V Stochastic Differential Equations XX,(1) = I +l t 'ijcr(XX )Xx,(1) dZ .t d s- s- s o We next observe that the components of Fk ,i(X(1), ... , x(k-i+l)) are sums of terms of the form k-i+1 Qj II II j=1 r=1 where L jexj = k, j (**) and where X(j),l is the l-th component of xU) E IRdj , and an "empty" product equals l. We thus want now to prove that E{sup 11t 'ijicr(X,X)F ·(XX,(1) ... Xx,(k-i+ 1))dZ /p} < Ks- N,z s-' , s- s_ t 0 for all i = 2, ... , N and for some constant K = K(k,p, cr, 'TIkp) (in the remain- der of the proof K = K(k,p, cr, 'TIkp) varies from line to line). And of course, it is enough to prove that if G is any monomial as in (**), then We will use Theorem 66 and the fact that 'iji cr is bounded. Note that Theo- rem 66 implies that for 2 :s: p' :s: p, where the constant K(p, 'TIp) depends only on p, 'TIp, and the dimensions of Hand Z. For this we see that the left side of the previous equation is smaller than k+1-i K E{ II sup IX:,(j) IPQj} :s: K j=1 s k+1-iII (E{sup IX~x,(j) Ikpfj} )jQj/k j=1 s by Holder's inequality, since L:j jexj = k. The recurrence assumption yields that each expectation above is smaller than some constant K(p, k, cr, 'TIkp) , and we obtain the required inequality. 0 We now turn to the positivity of solutions of stochastic differential equa- tions. This issue arises often in applications (for example in mathematical finance) when one wants to model a dynamic phenomenon where for reasons relating to the application, the solution should always remain positive. We begin with a very simple result, already in widespread use. 11 Eclectic Useful Results on Stochastic Differential Equations 345 Theorem 71. Let Z be a continuous semimartingale, cr a continuous func- tion, and suppose there exists a (path-by-path) unique, non-exploding solution to the equation Xt= Xo + it cr(Xs)XsdZs with Xo > 0 almost surely. Let T = inf{t > 0 : Xt = O}. Then P(T < (0) = O. In words, if the coefficient of the equation is of the form xcr(x) with cr(x) continuous on [0, (0), then the solution X stays strictly positive in finite time if it begins with a strictly positive initial condition. We also remark that in the applied literature the equation is often written in the form Note that if cr(x) is any function, one can then write cr(x) = x(O"~x)), but we then need the new function O"~x) to be continuous, well-defined for all nonnegative x, and we also need for a unique solution to exist that has no explosions. This of course amounts to a restriction on the function cr. Since xcr(x) being Lipschitz continuous is a sufficient condition to have a unique and non-exploding solution, we can require in turn for cr to be both bounded and Lipschitz, which of course implies that xcr(x) is itself Lipschitz. Proof of Theorem 71. Define Tn = inf{t > 0 : X t = lin or X t = X o V n}, and note that P(Tn > 0) = 1 because P(Xo > 0) = 1 and Z is continuous. Using Ito's formula up to time Tn we have and since cr is assumed continuous, it is bounded on the compact set [lin, nJ, say by a constant c. Since the stopping times Tn i T we see that on the event {T < oo} the left side of the above equation tends to 00 while the right side remains finite, a contradiction. Therefore P(T < (0) = O. 0 Theorem 71 has an analogue when Z is no longer assumed to be continuous. Theorem 72. Let Z be a semimartingale and let cr be a bounded, continuous function such that there exists a unique non-exploding solution of Xt= Xo +it cr(Xs-)Xs_dZs with Xo > 0 almost surely. If in addition we have I~Zs I ~ 11~1i: almost surely for all s 2: 0 for some c with 0 < c < 1, then the solution X stays positive a.s. for all t > O. 346 V Stochastic Differential Equations Proof II cr 1100 refers to the LOO norm of cr. The proof is similar to the proof of Theorem 71. Using Ito's formula gives i t litIn(IXtl) = In(Xo) + cr(Xs-)dZs - - cr(Xs_)2d[Z, Z]~o 2 0 +L {In IXsl-ln IXs-l- cr(Xs-)~Zs}, ss,t and let us consider the last term, the summation. We have L {In IXsl-ln IXs-l- cr(Xs-)~Zs} ss,t where IHsl ::s: 1 by the Mean Value Theorem, applied w-by-w, which implies the above is 1,,1 2 2 ::::: "2 L.J c:2cr(Xs-) (~Zs) , ss,t which is a convergent series for each finite time t. The only ways X can be zero or negative is to cross 0 while continuous, which cannot happen by an argument analogous to the proof of Theorem 71, or to jump to 0 or across into negative values. Assume X has not yet reached (-00,0] at time S-. Then we have I~Xsl ::s: Xs-lcr(Xs-)~Zsl ::s: X s-(l- c:) < X s-, which implies that X cannot cross into (-00, OJ by jumping. Hence we have that X must be positive on [0,(0). 0 We now turn to something completely different. In Sect. 6, Theorem 32, we showed that an equation of the form with Z a vector of independent Levy processes, had a (unique) solution which was strong Markov. We now show what amounts to a converse, which is per- haps surprising. That is, if one wants a solution of a stochastic differential equation to be time homogeneous Markov, one essentially must have that the driving semimartingale is a Levy process! Bibliographic Notes 347 Theorem 73. Let (0, F, IF, P) be a filtered probability space with Z a cadlag semimartingale defined on the space. Let f be a Borel function which is never o and is such that for every x E JR the equation has unique (strong) solution, which we denote Xx. If each of the processes Xx is a time homogeneous Markov process with the same transition semigroup, then Z is a Levy process. For the proof of this theorem, we assume the reader is familiar with the abstract theory of Markov processes, as can be found (for example) in [215]. Proof Let 0' be the function space of dtdHtg functions defined on JR+ with Xt(w) = w(t), the projection operator. Let IF' be the canonical filtration and let (en be the shift operators. If P~ denotes the law of Xx, then we have that (0' ,IF' , ei, X', PD is a Markov process in the Dynkin (or Blumenthal-Getoor) sense. Since f is never zero, we can write We can therefore define on 0', relative to each law P~, the stochastic integral Z' = it f(X' )-ldX't s- s' o and we have that the law of Z' under P~ is the law of the process Z - Zo. However Z' is also an additive functional, and hence the Markov property implies where we have also used that the law of Z' under P~ is the law of the process Z - Zoo This in turn implies that Z~+s - Z~ is P~ independent of F£, and hence also independent from Z~ for all r ::::: t, and it also implies that the law of Z~+s - Z~ is the same as the law of Zs - Zoo Using the identity of the laws of Z' under P~ with that of the process Z - Zo once again, the result follows. 0 Bibliographic Notes The extension of the HP norm from martingales to semimartingales was im- plicit in Protter [197fand first formally proposed by Emery [64J. A compre- hensive account of this important norm for semimartingales can be found in 348 V Stochastic Differential Equations Dellacherie-Meyer [46]. Emery's inequalities (Theorem 3) were first established in Emery [64], and later extended by Meyer [174]. Existence and uniqueness of solutions of stochastic differential equations driven by general semimartingales was first established by Doleans-Dade [51] and Protter [198], building on the result for continuous semimartingales in Protter [197J. Before this Kazamaki [123] published a preliminary result, and of course the literature on stochastic differential equations driven by Brownian motion and Lebesgue measure, as well as Poisson processes, was extensive. See, for example, the book of Gihman-Skorohod [81J in this regard. These results were improved and simplified by Doleans-Dade-Meyer [54J and Emery [65J; our approach is inspired by Emery [65]. Metiver-Pellaumail [161] have an alternative approach. See also Metivier [158J. Other treatments can be found in Doleans-Dade [52] and Jacod [103]. The approach of L. Schwartz [214] is nicely presented in [44J. The stability theory is due to Protter [199], Emery [65], and also to Metivier-Pellaumail [162J. The semimartingale topology is due to Emery [66] and Metivier-Pellaumail [162], while a pedagogic treatment is in Dellacherie- Meyer [46]. The generalization of Fisk-Stratonovich integrals to semimartingales is due to Meyer [171J. The treatment here of Fisk-Stratonovich differential equations is new. The idea of quadratic variation is due to Wiener [230J. Theorem 18, which is a random Ito's formula, appears in this form for the first time. It has an antecedent in Doss-Lenglart [58J, and for a very general version (containing some quite interesting consequences), see Sznit- man [222]. Theorem 19 generalizes a result of Meyer [171], and Theorem 22 extends a result of Doss-Lenglart [58]. Theorem 24 and its corollary is from Ito [99]. Theorem 25 is inspired by the work of Doss [57] (see also Ikeda- Watanabe [92J and Sussman [221]). The treatment of approximations of the Fisk-Stratonovich integrals was inspired by Yor [238J. For an interesting ap- plication see Rootzen [211]. For more examples of solutions of stochastic dif- ferential equations with formulas, see [128J. The results of Sect. 6 are taken from Protter [196J and Qinlar-Jacod- Protter-Sharpe [34]. A comprehensive pedagogic treatment when the Markov solutions are diffusions can be found in either Stroock-Varadhan [220] or Rogers-Williams [209, 21OJ. Work on flows of stochastic differential equations goes back to 1961 and the work of Blagovescenskii-Freidlin [18J who considered the Brownian case. For recent work on flows of stochastic differential equations, see Kunita [131, 133]' Ikeda-Watanabe [92] and the references therein. There are also flows results for the Brownian case in Gihman-Skorohod [81]' but they are L 2 rather than al- most sure results. Much of our treatment is inspired by the work of Meyer [177] and that of Uppman [224, 225] for the continuous case, however results are taken from other articles as well. For example, the example following The- orem 38 is due to Leandre [140], while the proof of Theorem 41, the non- confluence of solutions in the continuous case, is due to Emery [68]; an alter- Exercises for Chapter V 349 native proof is in Uppman [225J. For the general (right continuous) case, we follow the work of Leandre [141J. A similar result was obtained by Fujiwara- Kunita [78J. Our presentation on moment estimates follows [205J and [105], although it dates back to [16]. A perhaps simpler treatment using random measures can be found in [105]. See also [106J. The results on when solutions of stochastic differential equations are always positive is well known, at least in the Brown- ian case, but a reference is not easily found in the literature. The general case may be new. Theorem 73 is from [108J. Exercises for Chapter V Exercise 1. Let X be a solution of the stochastic differential equation dXt = X;dBt + X(dt with initial condition X o i= O. Show that X is defined on [0, T), where T = inf{t > 0 : Bt = l/Xo}. Moreover show that X is the unique solution on [0, T). Finally show that X explodes at T. Exercise 2. Let a(x) = JiXT. Note that a is Lipschitz continuous everywhere except at x = O. Let X be a solution of the equation with initial condition X o. Find an explicit solution of this equation and de- termine where it is defined. Exercise 3. Let X be the unique solution of the equation with X o given, and a and b both Lipschitz continuous. Let the generator of the (time homogeneous) Markov process X be given by Af(x) = lim Ptf(x) - f(x) t .....o,t> 0 t where Ptf(x) equals the transition function22 Ps(Xo, f) when X o = x almost surely. A is called the infinitesimal generator of X. Use Ito's formula to show that if f has two bounded continuous derivatives (written f E C~), then the limit exists and A is defined. In this case j is said to be in the domain of the generator A. 22 This notation is defined in Sect. 5 of Chap. I on page 35. 350 Exercises for Chapter V Exercise 4. (Continuation of Exercise 3.) With the notation and terminology of Exercise 3, show that the infinitesimal generator of X is a linear partial differential operator for functions I E C~ given by Exercise 5. Let I E Ct, let (j and b be Lipschitz continuous, let X be the unique solution of the stochastic integral equation and let A denote its infinitesimal generator. If X o = x, where x E JR, prove Dynkin's expectation formula for Markov processes in this case: Exercise 6. As a continuation of Exercise 5, show that (Pd)(y) - I(y) -It (PsAf)(y)ds = 0 for y E JR, and that lims!o PsAI = AI for (for example) all I in Coo with compact support. Exercise 7. As in Exercise 5, extend Dynkin's expectation formula by show- ing that it still holds if one replaces in equation (**) the integration upper limit t with a bounded stopping time T, and use this to verify Dynkin's for- mula in this case. That is, show that if I E CC and (j and b are both Lipschitz, and X is a solution of equation (*) of Exercise 5, then I(XT) - I(Xo) -iT AI(Xs)ds is a local martingale. Exercise 8. Let Z be a Levy process with Levy measure IJ and let X satisfy the equation X t = x +I t I(Xs-)dZs where I is Lipschitz continuous. The process X is taken to be d-dimensional, and Z is n-dimensional, so I takes its values in JRd X JRn. We use (column) vector and matrix notation. The initial value is some given x E JRd. Let px denote the corresponding law of X starting at x. Show that if 9 E Coo with compact support on JRd, then Exercises for Chapter V 351 d ("'2 )1 u 9 ..Ag(x) ='Vg(x)f(x)b + 2.2.: &xi&x j (x) (f(x)cf(x)*)'J ',J=l +Jv(dy) (g(x + f(x)y) - g(x) - 'Vg(x)f(x)) , where 'V9 is a row vector and b is the drift coefficient of Z, * denotes transpose, and A is of course the infinitesimal generator of X. Exercise 9. In the setting of Exercise 8 show that g(Xt ) - g(Xo) - it Ag(Xs)ds is a local martingale for each law px. Show further that it is a local martingale for initial laws of the form pJ.1., where PJ.1. is given by: for A C D measurable, PJ.1.(A) = JIRd PX(A)p,(dx) for any probability law p, on JRd. Exercise 10. Let Z be an arbitrary semimartingale, and let To = 0 and (Tn )n2: 1 denote the increasing sequence of stopping times of the jump times of Z when the corresponding jump size is larger than or equal to 1 in magnitude. Define En = sign(6.ZTJ with EO = 1. For a semimartingale Y define Un(Y)t to be the solution of the exponential equation l't = 1 + r l!;,_dYsJTn for Tn :s: t < Tn+1. Show that the (unique) solution of the equation Xt= 1+ it IXs-ldZs is given by X t = L:n>oXr1[Tn,Tn+d(t), where Xr = XTnUn(EnZ)t, for Tn:S: t < Tn+!' - Exercise 11. Show that if M is an L 2 martingale with 6.Ms > -1, and (M, M)oo is bounded, then £(M)oo > 0 a.s. and it is square integrable. (See [147].) Exercise 12. Let M be a martingale which is of integrable variation. Suppose also that the compensator of the process L:s 352 Exercises for Chapter V *Exercise 14 (Memin's criterion for exponential martingales). Let M be a local martingale, let J t = Lss;t ~Ms1{I.6.Msl2:~}'and let L t = J t - h and Nt = M t - Lt. (Both Land N are of course also local martingales.) Let At = [M,M]C + L I~Lsl + L (~Ns)2 O Exercises for Chapter V 353 Exercise 20. Use Exercises 16 through 19 as needed to show, via a Picard iteration method, that a unique solution exists to the stochastic differential equation where f is Lipschitz continuous and Z is an arbitrary semimartingale. Exercise 21. Show that there are an infinite number of solutions (on the same probability space) to the equation {t 1 t 2 X t = 3 Jo Xl ds + 3 Jo Xl dBs, Xo =0, where B is a standard Brownian motion. (See [121, page 293].) *Exercise 22. Let a satisfy a weakening of the Lipschitz condition of the form la(y) - a(x)1 :s: 1\;(lx - yl) for all x, y E JR, with J; K,(~)2du = 00 and I\; increasing with 1\;(0) = 0. (Such a condition is a variant of a standard condition for uniqueness of solutions in the theory of ordinary differential equations; see for example [35, page 60]. Let b be Lipschitz continuous. Show that there exists a unique solution of the equation where B is standard Brownian motion. (See [232].) Exercise 23. Let a and b be Lipschitz continuous and let X be the unique solution of equation (*) of Exercise 5. Show that 2xb(x) + a(x)2 :s: c + klxl 2 , where c and k are positive constants. Use Ito's formula on xl to show the following moment estimate: E{Xl} :s: (E{X5} + ct)ekt . (See [135].) *Exercise 24 (Euler method of approximation). Let a and b be con- tinuous and such that a unique path-by-path solution X of equation (*) of Exercise 5 exists. Let (~k)k>l be i.i.d. with mean zero and variance T 2 . Let Xo be independent of (~k)k~l and let X k be defined inductively by X k+1 = X k + a ~k+l + b(Xk ) . .;n n Define Xn(t) = X[ntj where [nt] denotes the integer part of nt. Let Bn(t) = In Lt~jl ~k, Vn(t) = [r;:], and finally observe that By Donsker's Theorem we know that (Bn , Vn ) converges weakly (in distri- bution) to (TB, V), where B is a standard Brownian motion and V(t) = t. Show that B n is a martingale, and that X n converges weakly to X. (See [136] and [137].) 354 Exercises for Chapter V *Exercise 25. Let Z be a Levy process which is a square integrable martin- gale. Let f be continuously differentiable with a bounded derivative and let Xx be the unique solution of Show that Xx is also a square integrable martingale. *Exercise 26. In the framework of Exercise 25, let X;x be the solution of Show that X' x is also a square integrable martingale. **Exercise 27. In the framework of Exercises 25 and 26, for each measurable function 9 with at most linear growth, Set Assume that f is infinitely differentiable, and that 9 is twice differentiable, bounded and both of its first two derivatives are bounded. Show that the function (t, x) f-> Ptg (x) is twice differentiable in x and onCe differentiable in t, that all the partial derivatives are continuous in (t, x), and further that VI Expansion of Filtrations 1 Introduction By an expansion of the filtration, we mean that we enlarge the filtration (Ftk::o to get another filtration (Htk:.o such that the new filtration satisfies the usual hypotheses and F t C H t , each t ?: o. There are three questions we wish to address: (1) when does a specific, given semimartingale remain a semi- martingale in the enlarged filtration; (2) when do all semimartingales remain semimartingales in the enlarged filtration; (3) what is a new decomposition of the semimartingale for the new filtration. The subject of the expansion of filtrations began with a seminal paper of K. Ito in 1976 (published in [100] in 1978), when he showed that if B is a standard Brownian motion, then one can expand the natural filtration (Ftk::o of B by adding the a-algebra generated by the random variable B1 to all Ft of the filtration, including of course Fo. He showed that B remains a semimartingale for the expanded filtration, he calculated its decomposition explicitly, and he showed that one has the intuitive formula where the integral on the left is computed with the original filtration, and the integral on the right is computed using the expanded filtration. Obviously such a result is of interest only for 0 ::::: t ::::: 1. We will establish this formula more generally for Levy processes in Sect. 2. The second advance for the theory of the expansion of filtrations was the 1978 paper of M. Barlow [7] where he considered the problem that if L is a positive random variable, and one expands the filtration in a minimal way to make L a stopping time, what conditions ensure that semimartingales remain semimartingales for the expanded filtration? This type of question is called progressive expansion and it is the topic of Sect. 3. 356 VI Expansion of Filtrations 2 Initial Expansions Throughout this section we assume given an underlying filtered probability space (n, F, (Ftk:o, P) which satisfies the usual hypotheses. As in previous chapters, for convenience we denote the filtration (Ft )t2:o by the symbol JF. The most elementary result on the expansion of filtrations is due to Jacod and was established in Chap. II (Theorem 5). We recall it here. Theorem 1 (Jacod's Countable Expansion). Let A be a collection of events in F such that if An, A,B E A then An n A,B = ¢, a: i- (3. Let H t be the filtration generated by Ft and A. Then every ((Ftk::o, P) semimartingale is an ((Ht )t2:o, P) semimartingale also. We also record a trivial observation as a second elementary theorem. Theorem 2. Let X be a semimartingale with decomposition X = M + A and let Q be a a-algebra independent of the local martingale term M. Let lHI denote the filtration obtained by expanding IF with the one a-algebra Q (that is, Ht = Ft V Q, each t ~ 0 and Ht = Ht+). Then X is an lHI semimartingale with the same decomposition. Proof Since the local martingale M remains a local martingale under lHI, the theorem follows. 0 We now turn to Levy processes and an extension of Ito's first theorem. Let Z be a given Levy process on our underlying space, and define lHI = (Htk,::o to be the smallest filtration satisfying the usual hypotheses, such that Zl is Ho measurable and Ft C Ht for all t ~ O. Theorem 3 (Ito's Theorem for Levy Processes). The Levy process Z is an lHI semimartingale. If moreover E {IZt I} < 00, all t ~ 0, then the process M t = Zt _ (till Zl - ZSds Jo 1- s is an lHI martingale on [0,00). Proof We begin by assuming E{ zl} < 00, each t > O. Without loss of gener- ality we can further assume E{ Ztl = O. Since Z has independent increments, we know Z is an IF martingale. Let 0 :s: s < t :s: 1 be rationals with s = j In and t = kin. We set Yi = Zi+l - Zi-. n n Then Zl - Zs = E~~/ Yi and Zt - Zs = E:~/ Yi. The random variables Yi are i.i.d. and integrable. Therefore 2 Initial Expansions 357 k-l n-l . n-l E{Zt - ZslZI - Zs} = E{L Yil L Yi} = k =J. L Yi i=j i=j n J i=j t-s = 1 _ S (ZI - Zs). The independence of the increments of Z yields E{Zt - Zsl1t s} = E{Zt - ZslZI - Zs}; therefore E{Zt - Zsl1ts} = ~=~(ZI - Zs) for all rationals, 0 ~ s < t ~ 1. Since Z is an iF martingale, the random variables (ZdO~t9 are uniformly integrable, and since the paths of Z are right continuous, we deduce E{Zt - Zsl1t s} = ~=~(ZI - Zs) for all reals, 0 ~ s < t ~ 1. By Fubini's Theorem for conditional expectations the above gives There is a potential problem at t = 1 because of the possibility of an explo- sion. Indeed this is typical of initial enlargements. However if we can show E{fol IZr::;slds} < 00 this will suffice to rule out explosions. By the sta- tionarity and independence of the increments of Z we have E{IZI - Zsl} ~ 2 1 1E{(ZI - Zs) } 2" ~ a(l - s) 2" for some constant a and for all s, 0 ~ s ~ 1. Therefore E{fol IZ~::::;slds} ~ a fo1~ds < 00. Note that if t > 1 then Ft = 1tt , and it follows that M is a martingale. Since Zt = Mt + fo1 zr::;s ds we have that Z is a semimartingale. Next suppose only that E{IZtl} < 00, t ~ 0 instead of Z being in L 2 as we assumed earlier. We define Jf = L .6.Zs 1{.6.Zs >l} O 358 VI Expansion of Filtrations therefore constant in a (random) neighborhood of 1, which in turn yields that 1 IJi_Jil . t J'_Ji .fo ~ds < 00 a.s., shows that (J't- fo ~dsk:?:o IS a local martingale for , '1 IJi-JillHl. Moreover it is a martingale for lHl as soon as E{fo ~ds} < 00. But the function t f-t E{Jf} = ait for all i by the stationarity of the increments. Hence Since Y, J 1 , and J2 are all independent, we conclude that M is an lHl' martin- gale. Since M is adapted to lHl, by Stricker's Theorem it is also an lHl martingale, and thus Z is an lHl semimartingale. Finally we drop all integrability assumptions on Z. We let Jl = 2:: .6.Zs 1{1.6.Zs l>1} and also X t = Zt - Jl. O 2 Initial Expansions 359 where since the random variable B l is 110 measurable, it can be moved inside the stochastic integral. We can extend this theorem with a simple iteration; we omit the fairly obvious proof. Corollary. Let Z be a given Levy process with respect to a filtration iF, and let 0 = to < tl < ... < tn < 00. Let lHI denote the smallest filtration satis- fying the usual hypotheses containing IF and such that the random variables Ztl' ... ,Ztn are all 110 measurable. Then Z is an lHI semimartingale. If we have a countable sequence 0 = to < it < ... < tn < ... , we let T = sUPn tn, with lHI the corresponding filtration. Then Z is an lHI semimartingale on [0, T). We next give a general criterion (Theorem 5) to have a local martingale remain a semimartingale in an expanded filtration. (Note that a finite varia- tion process automatically remains one in the larger filtration, so the whole issue is what happens to the local martingales.) We then combine this theo- rem with a lemma due to Jeulin to show how one can expand the Brownian filtration. Before we begin let us recall that a process X is locally integrable if there exist a sequence of stopping times (Tn)n>l increasing to 00 a.s. such that E{IXTnl{Tn>O}I} < 00 for each n. Of course, if Xo = 0 this reduces to the condition E{IXTnl} < 00 for each n. Theorem 4. Let M be an IF local martingale and suppose M is a semimartin- gale in an expanded filtration lHI. Then M is a special semimartingale in lHI. Proof. First recall that any local martingale is a special semimartingale. In particular the process Mt = sUPs 360 VI Expansion of Filtrations To remove the assumption EU;" H;d[M, M]s} < 00, we only need to recall that H . M is assumed to be locally square integrable, and thus take M stopped at a stopping time Tn that makes H . M square integrable, and we are reduced to the previous case. 0 Theorem 6 (Jeulin's Lemma). Let R be a positive, measurable stochas- tic process. Suppose for almost all s, R s is independent of Fsi and the law (or distribution) of R s is J.l which is independent of s, with J.l( {o}) = °and f;" xJ.l(dx) < 00. If a is a positive predictable process with f~ asds < 00 a.s. for each t, then the two sets below are equal almost surely: {l°° Rsasds < oo} = {l°° asds < oo} a.s. Proof We first show that U;" Rsasds < oo} c U;" asds < oo} a.s. Let A be an event with P(A) > 0, and let J = lA, Jt = E{JIFt}, the cadlag version of the martingale. Let j = inft Jt . Then j > °on {J = 1}. We have a.s. Consider next by Jensen's inequality, and this is equal to (E{lAIFt} - J.l(O,u])+, where J.l is the law of Rt . Continuing with (*) we have where nP(An ) ~ E{lAn 100 Rsasds} = E{l°° 1An R sasds} = E{l°° E{lAn R s IFs}asds} ~ E{l°° 2 Initial Expansions 361 {f;' asds < oo} a.s. We next show the inverse inclusion. Let Tn = inf{t > 0: f~ asds > n}. Then Tn is a stopping time, and n ~ E{f:n asds}. Moreover where 0: is the expectation of Rs , which is finite and constant by hypothesis. Therefore foTn Rsasds < 00 a.s. Let w be in {f;' asds < oo}. Then there exists an n (depending on w) such that Tn(w) = 00. Therefore we have the inclusion U;' Rsasds < oo} :J U;' asds < oo} a.s. and the proof is complete. 0 We are now ready to study the Brownian case. The next theorem gives the main result. Theorem 7. Let M be a local martingale defined on the standard space of canonical Brownian motion. Let JH[ be the minimal expanded filtration con- taining B 1 and satisfying the usual hypotheses. Then M is an JH[ semimartin- gale if and only if the integral f; vi=sld[M, B]sl < 00 a.s. In this case M t - f~i\1 Bl=~' d[M, B]s is an JH[ local martingale. Proof. By the Martingale Representation Theorem we have that every iF lo- cal martingale M has a representation M t = M o + f~ HsdBs, where H is predictable and f; H;ds < 00 a.s., each t > O. By Theorem 5 we know that M is an JH[ semimartingale if and only if f; IHsIIB~::::~slds is finite O < t < 1 Wi tak - IHsl d R - 1 IB1-Bsl Tha.s., _ _ . e e as - vr=s' an s - {s 362 VI Expansion of Filtrations Then H is trivially predictable and also f01H;ds < 00. However f01H s J=sds is divergent. Therefore M = H· B is an IF local martingale which is not an lHI semimartingale, by Theorem 7, where of course lHI = lHI(Bd. Thus we conclude that not all IF local martingales (and hence a fortiori not all semimartingales) remain semimartingales in the lHI filtration. We now turn to a general criterion that allows the expansion of filtration such that all semimartingales remain semimartingales in the expanded filtra- tion. It is due to Jacod, and it is Theorem 10. The idea is surprisingly simple: recall that for a cadlag adapted process X to be a semimartingale, if H n is a sequence of simple predictable processes tending uniformly in (t,w) to zero, then we must have also that the stochastic integrals H n . X tend to zero in probability. If we expand the filtration by adding a a-algebra generated by a random variable L to the IF filtration at time 0 (that is, a {L} is added to Fo), then we obtain more simple predictable processes, and it is harder for X to stay a semimartingale. We will find a simple condition on the random variable L which ensures that this condition is not violated. This approach is inherently simpler than trying to show there is a new decomposition in the expanded filtration. We assume that L is an (JE, f)-valued random variable, where E is a stan- dard Borel space1 and £ are its Borel sets, and we let lHI(L) denote the smallest filtration satisfying the usual hypotheses and containing both L and the orig- inal filtration IF. When there is no possibility of confusion, we will write lHI in place of lHI(L). Note that if Y E H~ = F t V a{L}, then Y can be written Y(w) = G(w,L(w)), where (w,x) f-+ G(w,x) is an F t 0£ measurable function. We next recall two standard theorems from elementary probability theory. Theorem 8. Let xn be a sequence of real-valued random variables. Then X n converges to 0 in probability if and only if limn-->oo E{min(l, Ixnl)} = O. A proof of Theorem 8 can be found in textbooks on probability (see for example [109]). We write 1/\ IXnl for min(l, IXnl). Also, given a random variable L, we let Qt(w, dx) denote the regular conditional distribution of L with respect to Ft , each t ~ O. That is, for any A E £ fixed, Qt(·, A) is a version of E{l{LEA}IFt }, and for any fixed w, Qt(w, dx) is a probability on £. A second standard elementary result is the following. Theorem 9. Let L be a random variable with values in a standard Borel space. Then there exists a regular conditional distribution Qt(w, dx) which is a version of E{l{LEdx} 1Ft}. For a proof of Theorem 9 the reader can see, for example, Breiman [23, page 79]. 1 (lE, E) is a standard Borel space if there is a set rEB, where B are the Borel subsets of JR, and an injective mapping r such that 2 Initial Expansions 363 Theorem 10 (Jacod's Criterion). Let L be a random variable with values in a standard Borel space (JE,£), and let Qt(w,dx) denote the regular condi- tional distribution of L given Ft , each t ~ 0. Suppose that for each t there exists a positive (J -finite measure "It on (JE, £) such that Qt (w, dx) « "It (dx) a.s. Then every IF semimartingale X is also an lHl(L) semimartingale. Proof. Without loss of generality we assume Qt (w, dx) « "It (dx) surely. Then by Doob's Theorem on the disintegration of measures there exists an £ 0 Ft measurable function qt(x,w) such that Qt(w,dx) = qt(x,w)"It(dx). Moreover since E{fJE Qt(-, dx)} = E{E{l{YEJE} IFd} = P(Y E JE) = 1, we have E{L Qt(-,dx)} = E{L qt(x,w)"It(dx)} = LE{qt(x,w)}"It(dx) = 1. Hence for almost all x (under TJt(dx)), we have qt(x,') E L1(dP). Let X be an IF semimartingale, and suppose that X is not an lHl( L) semi- martingale. Then there must exist au> °and an c > 0, and a sequence Hn of simple predictable processes for the lHl filtration, tending uniformly to °but such that infn E{l /\ IHn . XI} ~ c. Let us suppose that tn ~ u, and n-l Hf = L J;'l(ti,tHd(t) i=O with J[' EFt, Va{L}. Hence J[' has the form gi(W, L(w)), where (w, x) f-+ gi(W, x) is Fti 0 £ measurable. Since H n is tending uniformly to 0, we can take without loss IHnl ~ lin, and thus we can also assume that Igil ~ lin. We write n-l H~'X(w) = L gi(W, x)l(ti,ti+l](t), i=O and therefore (x, w) f-+ H~'x(w) and (x, w) f-+ (Hn,x . X)t(w) are each £ 0 Fu measurable, °~ t ~ u. Moreover one has dearly Hn·X = Hn,L·X. Combining the preceding, we have E{l/\ IHn . Xul} = E{h (1/\ IHn,x . Xu I)Qu (-, dx)} = E{h (1/\ IHn,x . Xu I)qu (-, dX)TJu(dx)} = hE{(l /\ IHn,x. X ul)qu(-,dx)}"Iu(dx) where we have used Fubini's Theorem to obtain the last equality. However the function hn(x) = E{(l /\ IHn,x . Xul)qu(-, x)} ~ E{qu(-, x)} E U(dTJu), and since hn is non-negative, we have 364 VI Expansion of Filtrations lim E{l/\ IHn . Xul} = lim rE{(l /\ IHn . X ul)quLx)}7]u(dx) n---+oo n---+oo lIE = r lim E{(l/\ IHn . X ul)qu(-,x)}7]u(dx) (*)lIE n-H)O by Lebesgue's Dominated Convergence Theorem. However quLx) E L 1(dP) for a.a. x (under (d7]u)), and if we define dR = cquL x)dP to be another proba- bility, then convergence in P-probability implies convergence in R-probability, since R« P. Therefore limn.-oo ER{(l /\ IHn,x. Xul)} = °as well, which im- plies . 10= hm -ER{(l/\ IHn,x . Xul)} n---+oo c = Ep{(l /\ IHn,x. Xu I)qu (-, x)} for a.a. x (under (d7]u)) such that quL x) E Ll(dP). Therefore the limit of the integrand in (*) is zero for a.a. x (under (d7]u)), and we conclude lim E{(l/\ IHn . Xul)} = 0, n.-oo which is a contradiction. Hence X must be a semimartingale for the filtration lHlo, where Hf = Ft V a {L}. Let X = M + A be a decomposition of X under lHlo. Since lHlo need not be right continuous, the local martingale M need not be right continuous. However if we define Mt = Mt if t is rational; and limu"t,uEIQ Mu if t is not rational; then Mt is a right continuous martingale for the filtration lHl where H t = nu>t H~. Letting At = Xt - Mt , we have that Xt = Mt +At is an lHl decomposition of X, and thus X is an lHl semimartingale. o A simple but useful refinement of Jacod's Theorem is the following where we are able to replace the family of measures TJt be a single measure 7]. Theorem 11. Let L be a random variable with values in a standard Borel space (JE, £), and let Qt (w, dx) denote the regular conditional distribution of L given Ft , each t ~ 0. Then there exists for each t a positive a-finite measure 7]t on (JE,£) such that Qt(w, dx) « 7]t(dx) a.s. if and only if there exists one positive a-finite measure 7](dx) such that Qt(w, dx) « 7](dx) for all w, each t > 0. In this case TJ can be taken to be the distribution of L. Proof. It suffices to show that the existence of TJt for each t > °implies the existence of TJ with the right properties; we will show that the dis- tribution measure of L is such an 7]. As in the proof of Theorem 10 let (x,w) 1--4 qt(x,w) be £0Ft measurable such that Qt(w,dx) = qt(x,w)TJt(dx). Let at(x) = E{qt(x, w)}, and define ( ) { q~(x(~w)), ifat(x) >0,rt x,w = t0, otherwise. 2 Initial Expansions 365 Note that at(x) = 0 implies qt(x,·) = 0 a.s. Hence, qt(x,w) = rt(x,w)at(x) a.s.; whence rt(x, w)at(x)ryt(dx) is also a version of Qt(w, dx). Let ry be the law of L. Then for every positive £ measurable function 9 we have Jg(x)ry(dx) = E{g(L)} = E{k g(x)Qt(-, dx)} = E{k g(x)qt(x, .)1]t(dx)} = k g(x)E{qt(x, ·)}ryt(dx) = 1g(x)at(x)ryt(dx) from which we conclude that at(x)ryt(dx) = ry(dx). Hence, Qt(w, dx) = Tt(w,x)TJ(dx), and the theorem is proved. 0 We are now able to re-prove some of the previous theorems, which can be seen as corollaries of Theorem 11. Corollary 1 (Independence). Let L be independent of the filtration IF. Then every IF semimartingale is also an lHI(L) semimartingale. Proof Since L is independent of Ft , E{g(L) 1Ft} = E{g(L)} for any bounded, Borel function g. Therefore E{g(L) 1Ft } = 1Qt(w,dx)g(x) = k ry(dx)g(x) = E{g(L)}, from which we deduce Qt(w, dx) = TJ(dx), and in particular Qt(w, dx) « TJ(dx) a.s., and the result follows from Theorem 11. 0 Corollary 2 (Countably-valued random variables). Let L be a random variable taking on only a countable number of values. Then every IF semi- martingale is also an lHI( L) semimartingale. Proof Let L take on the values (}:1, (}:2, (}:3, .... The distribution of L is given by ry(dx) = I:~1 P(L = (}:i)ca, (dx), where ca,(dx) denotes the point mass at (}:i. With the notation of Theorem 11, we have that the regular conditional distribution of L given :Ft , denoted Qt(w, dx), has density with respect to TJ given by The result now follows from Theorem 11. o 366 VI Expansion of Filtrations Corollary 3 (Jacod's Countable Expansion). Let A = (AI, A 2 , ... ) be a sequence of events such that Ai n Aj = 0, i =1= j, all in F, and such thatU:l Ai = D. Let lHI be the filtration generated by iF and A, and satisfying the usual hypotheses. Then every iF semimartingale is an lHI semimartingale. Proof. Define L = 2:::1 2-i I Ai . Then lHI = lHI(L) and we need only to apply the preceding corollary. 0 Next we consider several examples. Example (Ito's example). We first consider the original example of Ito, where in the standard Brownian case we expand the natural filtration iF with a{Bl }. We let lHI denote lHI(BI). We have where Tit (dx) is the law of Bl - Bt and where we have used that B l - Bt is in- dependent of Ft. Note that "It(dx) is a Gaussian distribution with mean 0 and variance (I-t) and thus has a density with respect to Lebesgue measure. Since Lebesgue measure is translation invariant this implies that Qt(w, dx) « dx a.s., each t < 1. However at time 1 we have E{g(BI)IFl } = g(BI), which yields Ql(W, dx) = C{B1 (w)} (dx), which is a.s. singular with respect to Lebesgue mea- sure. We conclude that any iF semimartingale is also an lHI(Bl ) semimartingale, for 0 ~ t < 1, but not necessarily including 1. This agrees with Theorem 7 which implies that there exist local martingales in iF which are not semimartin- gales in lHI(Bl ). Our next example shows how Jacod's criterion can be used to show a somewhat general, yet specific result on the expansion of filtrations. Example (Gaussian expansions). Let iF again be the standard Brownian filtration satisfying the usual hypotheses, with B a standard Brownian motion. Let V = I;' g(s)dBs, where Iooo g(s)2ds < 00, 9 a deterministic function. Let a = inf{t > 0: ftoo g(s)2ds = O}. If h is bounded Borel, then as in the previous example E{h(v)IFtl = E{h(l t g(s)dBs +100 g(s)dBs)IFtl = Jh(lt g(s)dBs + x) "It(dx), where "It is the law of the Gaussian random variable It' g(s)dBs. If a = 00, then "It is non-degenerate for each t, and "It of course has a density with respect to Lebesgue measure. Since Lebesgue measure is translation invariant, we conclude that the regular conditional distribution of Qt(w, dx) of V given Ft also has a density, because 2 Initial Expansions 367 Qt(W, h) = E{h(V)IFtl = Jh(it g(s)dBs + x)1Jt(dx). Hence by Theorem 10 we conclude that every IF semimartingale is an lHI(V) semimartingale. Example (expansion via the end of a stochastic differential equa- tion). Let B be a standard Brownian motion and let X be the unique solution of the stochastic differential equation X t = X o + it a(Xs)dBs + it b(Xs)ds where a and b are Lipschitz. In addition, assume a and b are chosen so that for h Borel and bounded, E{h(X1)IFtl = Jh(x)7r(1 - t, Xt, x)dx where 7r(1 - t,u,x) is a deterministic function. 2 Thus Qt(w,dx) = 7r(1 - t, Xt(w), x)dx, and Qt(w, dx) is a.s. absolutely continuous with respect to Lebesgue measure if t < 1. Hence if we expand the Brownian filtration IF by initially adding Xl> we have by Theorem 10 that every IF semimartingale is an lHI( Xl) semimartingale, for 0 ::; t < 1. The mirror of initial expansions is that of filtration shrinkage. This has not been studied to any serious extent. We include one result (Theorem 12 below), which can be thought of as a strengthening of Stricker's Theorem, from Chap. II. Recall that if X is a semimartingale for a filtration lHI, then it is also a semimartingale for any subfiltration G, provided X is adapted to G, by Stricker's Theorem. But what if a subfiltration IF is so small that X is not adapted to it? This is the problem we address. We will deal with the optional projection Z of X onto IF. Definition. Let H = (Ht)t>o be a bounded measurable process. It can be shown that there exists a unique optional process °H, also bounded, such that for any stopping time T one has E{HT1{T 368 VI Expansion of Filtrations Definition. LetH = (Ht)t>o be a bounded measurable process. It can be shown that there exists a predictable process PH, also bounded, such that for any predictable stopping time T one has The process PHis called the predictable projection of H. It follows that for the optional projection, for each stopping time T we have °HT = E{HTIFT} a.s. on {T < oo} whereas for the predictable projection we have that PHT = E{HTIFT-} a.s. on {T < oo} for any predictable stopping time T. For fixed times t > 0 we have then of course °Ht = E{HtIFd a.s., and one may wonder why we don't simply use the "process" (E{HtIFd )(.:0:0 instead of the more complicated object °H. The reason is that this is defined only almost surely for each t, and we have an uncountable number of null sets; therefore it does not uniquely define a process. (Related to this observation, note in contrast that in some cases (E{HtIFt})t>o might exist even when °H does not). We begin our treatment with two simple lemmas. Lemma. Let IF c 3 Progressive Expansions 369 The limitation of the two preceding lemmas is the need to require integra- bility of the random variables X t for each t ?: O. We can weaken this condition by a localization procedure. Definition. We say that a 370 VI Expansion of Filtrations Then L = sup{t : (t,w) E A}. In this sense every positive random variable is the end of a random set. Instead however let us begin with a random set A C JR+ x D and define L to be the end of the set A. That is, L(w) = sup{t: (t,w) E A} where we use the (unusual) convention that sup(0) = 0-, where {o-} is an extra isolated point added to the non-negative reals [O,ooJ and which can be thought of as 0- < O. We also define Fo- = Fo. The purpose of {O-} is to distinguish between the events {w : A(w) = 0} and {w : A(w) = {O}}, each of which could potentially be added to the expanded filtration. The smallest filtration expanding IF and making the random variable L a stopping time is t} = r t n {L > t}}. This filtration is easily seen to satisfy the usual hypotheses, and also it makes L into a stopping time. Thus 3 Progressive Expansions 371 and +00. We will denote A L = (Afk~_o, the (predictable) compensator of 1{t2L} for the filtration IF. Therefore if J is an IF predictable bounded process we have E{h1{L>O-}} = E{ r JsdA~}. l[O,ool We now define what will prove to be a process fundamental to our analy- sis. The process Z defined below was first used in this type of analysis by J. Azema [3]. Recall that if H is a (bounded, or integrable) IFL process, then its optional projection 0 H onto the filtration IF exists. We define Note that 1{L>t} is decreasing, hence by the lemma preceding Theorem 12 we have that Z is an IF supermartingale. We next prove a needed technical result. Theorem 13. The set {t : 0 :S t :S 00, Zt- = O} is contained in the set (L, 00] and is negligible for the measure dAL . Proof. Let T(w) = inf{t 2:: 0 : Zt(w) = 0, or Zt_(w) = 0 for t > O}. Then it is a classic result for supermartingales that for almost all w the function t I---t Zt(w) is null on [T(w), 00]. (This result is often referred to as "a non- negative supermartingale sticks at zero.") Thus we can write {Z = O} as the stochastic interval [T,oo], and on [0, T) we have Z > 0, Z_ > O. We have E{A~A~} = P(T < L) = E{ZT1{T 0, Z_ > 0 on [0, L). Next observe that the set {Z_ = O} is predictable, hence 0 = E{1{zt_=o}d1{L>t}} = E{1{zt_=o}dAf} and hence {Z_ = O} is negligible for dAL . Note that this further implies that P(ZL- > 0) = 1, and again that {Z_ = O} C (L,oo]. 0 We can now give a description of martingales for the filtration lFL , as long as we restrict our attention to processes stopped at the time L. What happens after L is more delicate. For an integrable process J we let P J denote its predictable projection. Theorem 14. Let Y be a random variable with E{IYI} < 00. A right con- tinuous version of the martingale Yl = E{YIFF} is given by the formula Moreover the left continuous version Y_ is given by Yl- = Z1 P(Y1 CO,LJ) + Y1 CL ,oo)' t- 372 VI Expansion of Filtrations Proof. Let OL denote the optional a-algebra on JR+ x D, corresponding to the filtration lFL. On [0, L), OL coincides with the trace of 0 on [0, L). (By o we mean of course the optional a-algebra on lR+ x D corresponding to the underlying filtration IF.) Moreover on [L,oo), OL coincides with the trace of the a-algebra B(lR+) 0Foo on [L, 00). The analogous description of pL holds, with [O,L) replaced by (O,L], and with [L,oo) replaced with (L,oo).1t is then simple to check that the formulas give the bona fide conditional expectations, and also the right continuity is easily checked on [0, L) and [L, 00) separately. The second statement follows since Z_ > °on (0, L] by Theorem 13. 0 We now make a simplifying assumption for the rest of this paragraph. This assumption is often satisfied in the cases of interesting examples, and it allows us to avoid having to introduce the dual optional projection of the measure cLl{L>o}, Simplifying assumption to hold for the rest of this paragraph. We assume L avoids all IF stopping times. That is, P(L = T) = °for all IF stopping times T. Definition. The martingale M L given by MF = Af + Zt is called the fun- damental L martingale. Note that it is trivial to check that M L is in fact a martingale, since AL is the compensator of 1 - Z. Note also that M{;, = A~, since Zoo = 0. Last, note that it is easy to check that M L is a square integrable martingale. Theorem 15. Let X be a square integrable martingale for the IF filtration. Then (Xt/\dt>o is a semimartingale for the filtration lFL. Moreover Xt/\L - f~/\L z~_ d(X,ML)s is a martingale in the lFL filtration. Proof. Let C be the (non-adapted) increasing process Ct = l{t~L}' Since C has only one jump at time L we have E{XL} = E{fooo XsdCs}. Since X is a martingale it jumps only at stopping times, hence it does not jump at L, and using that A L is predictable and hence natural we get E{XL} = E{l OO Xs_dCs} = E{l OO Xs_dA~} = E{XooA~} = E{XooM~} = E{[X, ML]oo} = E{ (X, ML)oo}. (*) Suppose that H is a predictable process for JFL, and J is a predictable process for IF which vanishes on {Z_ = O} and is such that J = H on (0, L]. We are assured such a process J exists by the lemma preceding Theorem 13. Suppose first that H has the simple form H = hl(t,oo) for bounded h E FF. If j is an Ft random variable equal to h on {t < L}, then we can take J = jl(t,oo) and we obtain H . X oo = h(XL - Xd1{t 3 Progressive Expansions 373 Since (X, M L) is IF predictable, we can replace H by Hi~,L) because it has the same predictable projection on the support of d(X, M L ). This yields 100 lL HE{ HsdXs} = E{ Z s d(X, ML)s}.o 0 s- Last if we take the bounded lFL predictable process H to be a stochastic interval [0, T 1\ L], where T is an lFL stopping time, we obtain E{XT/\L - f:/\L z~_ d(X,ML)s} = 0, which implies by Theorem 21 of Chap. I that XT/\L - f:/\L z~_ d(X, ML)s is a martingale. 0 We do not need the assumption that X is a square integrable martingale, which we made for convenience. In fact the conclusion of the th~orem holds even if X is only assumed to be a local martingale. We get our main result as a corollary to Theorem 15. Corollary. Let X be a semimartingale for the IF filtration. Then (Xt/\dt>o is a semimartingale for the filtration lFL . - Proof If X is a semimartingale then it has a decomposition X = M + D. The local martingale term M can be decomposed into X = V + N, where V and N are both local martingales, but V has paths of bounded variation on compact time sets, and N has bounded jumps. (This is the Fundamental Theorem of Local Martingales, Theorem 25 of Chap. III.) Clearly V and D remain finite variation processes in the expanded filtration lFL , and since M has bounded jumps it is locally bounded, hence locally square integrable, and since every IF stopping time remains a stopping time for the lF L filtration, the corollary follows from Theorem 15. 0 We need to add a restriction on the random variable L in order to study the evolution of semimartingales in the expanded filtration after the time L. Definition. A random variable L is called honest if for every t :S 00 there exists an F t measurable random variable L t such that L = L t on {L :S t}. Note that in particular if L is honest then it is F oo measurable. Also, any stopping time is honest, since then we can take L = L 1\ t which is of course Ft measurable by the stopping time property. Example. Let X be a bounded carllag adapted processes, and let xt* = sUPs 374 VI Expansion of Filtrations This is often described verbally by saying "L is honest if it is the end of an optional set." Proof. The end of an optional set is always an honest random variable. Indeed, on {L :'S t}, the random variable L coincides with the end of the set An([O, t] x 0), which is Ft measurable. For the converse we suppose L is honest. Let (Lth?o be an IF adapted process such that L = Lt on {L :'S t}. Since we can replace Lt with Lt 1\ t we can assume without loss of generality that L t :'S t. There is also no loss of generality to assume that L t is increasing with t ; since we can further replace Lt with sUPao is optional for the filtration IF. Last, L is now the end of the optional set {(t,w) : Lt(w) = t}. 0 When L is honest we can give a simple and elegant description of the filtration FL. Theorem 17. Let L be an honest time. Define gt = {r : r = (A n {L > t}) U (B n {L :'S t} ) for some A, B E Ft } Then 3 Progressive Expansions 375 U = H1[O,LJ + K1(L,ooj' One can extend this result to predictable processes by the Monotone Class Theorem. 0 Theorem 18. Let X be a square integrable martingale for the IF filtration. Then X is a semimartingale for the 376 VI Expansion of Filtrations Proof of Theorem 18. We first observe that without loss of generality we can assume X o = O. Let H be a bounded IF predictable process. We define stochas- tic integrals at the random time L by where H· X = (H· Xt)t>o is the IF stochastic integral process. That is, we use the usual definition of the stochastic integral, sampling it at the random time L. When H is a simple predictable process, these are reasonable definitions. Moreover we know from the lemma that if Hand J are both bounded IF pre- dictable processes and if also H = Jon (L, 00]' then f; HsdXs = f; JsdXs a.s., so the definition is well-defined. Since X is an IF martingale in L 2 with Xo = 0, we have E{fooo HsdXs} = O. Applying the equalities (*) on page 372 to H . X we have where the second equality uses (**) on page 373. Recalling E{fooo HsdXs} = 0 and combining this with the above gives (where we have changed the name of the process H to K for clarity slightly later in the proof) We next replace K with Kl{z_ 4 Time Reversal 377 where Hand K are IF predictable processes (not necessarily simple pre- dictable, however). If we further take U bounded by 1 and take the supre- mum Over all such simple predictable processes of the elementary 378 VI Expansion of Filtrations Ft = 11 V N, where N are the P-null sets of F1 , while P = :F6 V N is the analogous backward filtration for B. Our first two theorems are obvious, and we omit the proofs. Theorem 20. Let B be a standard Brownian motion on [0,1]. Then the time reversal B of B is also a Brownian motion with respect to its natural filtration. Theorem 21. Let Z be a Levy process on [0, 1]. Then Z is also a Levy process, with the same law as - Z, with respect to its natural filtration. In what follows let us assume that we are given a forward filtration IF = (Ft )O:st:S1 and a backward filtration G= Wt )O:st:S1. We further assume both filtrations satisfy the usual hypotheses. Definition. A cadlag process Y is called an (IF, G) reversible semimartin- gale if Y is an IF semimartingale on [0,1] and Y is a G semimartingale on [0,1). Note the small lack of symmetry; for the time reversed process we do not include the final time 1. This is due to the occurrence of singularities at the terminal time 1, and including it as a requirement would exclude interesting examples. Let Tt = (to, . .. ,tk) denote a partition of [0, t] with to = 0, tk = t, a :::; t :::; 1, and let Definition. Let H, Y be two cadlag processes. The quadratic covariation of Hand Y, denoted [H, Y], is defined to be lim Srn (H, Y) = [H, Y]t n--+ 4 Time Reversal 379 i = 2, ... ,n - 1. Note that one can always choose the partition points {sd in this manner because since for each w, S f---> Y" (w) has only countably many jumps, and hence Jo1 1{l ilYsl>0}ds = a a.s. But then Jo1 P(I~Ysl > O)ds = 0, which in turn implies that P(I~Ysl > 0) = a for almost all s in [0, 1J. We now define three new processes: n-2 AT = H(1-t)-~Y1-t + L H Si (y"i+l - Ys.) + H Sn _1 (Y1- - Y"n-J i=l n-1 BT = - L HSi+l-(YSi+l- - YSi -) i=l n-2 C T = H1-t~Y1-t + L {(HSi+1 - HS.)(YSi+1 - Ys.)} i=l +(H1- - Hsn_J(Y1- - YSn-J· Let Tn be a sequence of partitions of [0,1] with limn--->CXl mesh(Tn) = O. Then lim CTn = [H, Yh- - [H, Yh-t + ~H1-t~Y1-t n--->CXl = [H, Yh- - [H, Yl(1-t)- ---------- = -[H,Y]t Since Y is (IF, G) reversible by hypothesis, we know that CT is G adapted. Hence [H;'Yl is Gadapted and moreOVer since it has paths of finite variation by hypothesis, it is a semimartingale. Since H is cadlag we can approximate the stochastic integral with partial sums, and thus Since Y. - Y. = _(y1-si - y1-Si+l) we haveSi+l- Si- , 380 VI Expansion of Filtrations n-2 = Hl-tl:.Yl - t + L H Si+1(YSi+1 - Y;,,) + H l- (Yl - - YSn - 1) i=l n-l - L HSi+1(y;'i+1- - YSi -) i=l n-2 = H1-tl:.Yl- t +L H Si+1(l:.Y;,i+1 -l:.Y;,,) - l:.Hl (Yl - - YSn-J - H ll:.YSn_1· i=l Since we chose our partitions Tn with the property that l:.Y;,i = aa.s. for each partition point Si, the above simplifies to which tends to a since S 1 decreases to 1- t, and Sn-l increases to 1. Therefore ~stablishes the desired formula, and since we have already seen that [H, Y]t is a semimartingale, we have that X is also a reversible semimartingale as a consequence of the formula. 0 We remark that in the case where 17 is a 4 Time Reversal 381 We will show U is an (IF, JHr) reversible semimartingale. The hypotheses on f imply that its primitive F is the difference of two conVeX functions, and therefore M t = F(Bt ) - F(Bo) is an IF semimartingale. Moreover £:It = F(B1- t ) - F(B1), and since we already know that B 1- t is an 1HI semimartingale by Theorem 3, by the convexity of F we conclude that £:I is also an 1HI semimartingale. Therefore f~ f(Bs)dBs is an (IF, 1HI) reversible semi- martingale as soon as ! fIR LfT/(da) is one, where T/ is the signed measure 'sec- ond derivative' of F. Finally, what we want to show is that At = ~ fIR Lfp,(da) is an (IF, JHr) reversible semimartingale, for any signed measure p,. This will imply that U is an (IF, JHr) reversible semimartingale. Since A is continuous with paths of finite variation on [0,1], all we really need to show is that At E Ht , each t. But At = A 1- t - A 1 = flR(L~-t - L~)p,(da), and where Af is the local time at level X of the standard Brownian motion f3. Therefore At = - flR(A~-Bl )p,(da). Since Af is 1HI adapted, and since B 1 E H O, we conclude A is lHI adapted, and therefore U is an (IF, lHI) reversible semimartingale. We next consider the time reversal of stochastic differential equations, which is perhaps more interesting than the previous two examples. Let B be a standard Brownian motion and let X be the unique solution of the stochastic differential equation where 0" and b are Lipschitz, and moreover, 0" and b are chosen so that for h Borel and bounded, where 7f(1 - t, u, x) is a deterministic function. As in the example expansion via the end of a stochastic differential equation treated earlier on page 367, 382 VI Expansion of Filtrations we know (as a consequence of Theorem 10) that if IF is the natural, completed Brownian filtration, we can expand IF with Xl to get JH[ : Ht = nu>t Fu V O"{Xd, and then all IF semimartingales remain JH[ semimartingales on [0,1). We fix (t, w) and define ¢ : lR ----+ lR by ¢(x) = X(t, w, x) where X t X(t,w,x) is the unique solution of Xt = x +I t O"(Xs)dBs +I t b(Xs)ds for x E R Recall from Chap. V that ¢ is called the flow of the solution of the stochastic differential equation. Again we have seen in Chap. V that under our hypotheses on 0" and b the flow ¢ is injective. For a < s < t < 1 define the function ¢s,t to be the flow of the equation It now follows from the uniqueness of solutions that X t = ¢sAXs), and in particular Xl = ¢t,I(Xt), and therefore ¢~I(XI) = Xt, where ¢~1 is of course the inverse function of ¢s,t. Since the solution Xs,t of equation (*) is g = O"{Bv - B u ; s .::; u, v .::; t} measurable, we have ¢t,1 E F I- t , where P = O"{ih a .::; s .::; t} V N where N are the null sets of :F. Let H t = nu>t P V O"{Xd and we have fj is an JHr = (Ht)o$t$l semimartingale , and that Xt = ¢iUXI) is H I- t measurable. Therefore Xt E Ft and also at the same time X: E H I - t . Finally note that since X is a semimartingale and 0" is CI , the quadratic covariation [O"(X), B] exists and is of finite variation, and we are in a position to apply Theorem 22. Theorem 23. Let B be a standard Brownian motion and let X be the unique solution of the stochastic differential equation Xt = Xo +I t O"(Xs)dBs +I t b(Xs)ds for a .::; t;:; 1, where 0" and b are Lipschitz, and moreover, 0" and b are chosen so that for h Borel and bounded, E{h(XI)IFd = f h(x)'n-(l - t,Xt,x)dx where 7f(1 - t, u, x) is a deterministic function. Let JH[ be given by Ht = nu>t F u V O"{XI}. Then B is an (IF, JHr) reversible semimartingale, and X t = X I - t satisfies the following backward stochastic differential equation for a .::; t .::; 1. In particular, X is an (IF, JHr) reversible semimartingale. Bibliographic Notes 383 Proof We first note that B is an (IF,JHr) reversible semimartingale as we saw in the example on page 367. We have that [O"(X), B] = J~ O"'(Xs)O"(Xs)ds, which is clearly of finite variation, since 0"' is continuous, and thus O"'(X) has paths bounded by a random constant on [0, 1]. In the discussion preceding this theorem we further established that O"(Xt ) E 1i1- t , and of course a(Xt ) E :Ft. Therefore by Theorem 22 we have ~ 1 Observe that [O"(X) , B]t = J1- t O"'(Xs)O"(Xs)ds. Use the change of variable u = 1 - s in the preceding integral and also in the term ILt b(Xs)ds to get and the proof is complete. Bibliographic Notes o The theory of expansion of filtrations began with the work of K. Ito [100] for initial expansions, and with the works of M. Barlow [7] and M. Yor [239] for progressive expansions. Our treatment has benefited from some private notes P. A. Meyer shared with the author, as well as the pedagogic treatment found in [44]. A comprehensive treatment, including most of the early important results, is in the book of Th. Jeulin [114]. Excellent later summaries of main results, including many not covered here, can be found in the lecture notes volume edited by Th. Jeulin and M. Yor [117] and also in the little book by M. Yor [247]. Theorem 1 is due to J. Jacod, but we first learned of it through a paper of P. A. Meyer [175], while Theorem 3 (Ito's Theorem for Levy processes) was established in [107], with the help of T. Kurtz. Theorems 6 and 7 are of course due to Th. Jeulin (see page 44 of [114]); see also M. Yor [245]. Theorem 10, Jacod's criterion, and Theorem 11 are taken from [104]; Jacod's criterion is the best general result on initial expansions of filtrations. The small result on filtration shrinkage is new and is due to J. Jacod and P. Protter. Two seminal papers on progressive expansions after M. Barlow's original paper are those of Th. Jeulin and M. Yor [115] and [116]. The idea of using a slightly larger filtration than the minimal one when one progressively expands, dates to M. Yor [240], and the idea to use the process Zt =0 l{L>t} originates with J. Azema [3]. The lemma on local behavior of stochastic integrals at random times is due to C. Dellacherie and P. A. Meyer [46], and the idea of the proof of Theorem 18 is ascribed to C. Dellacherie in [44]. The results on time reversal are inspired by the work of J. Jacod and P. Protter [107] and 384 Exercises for Chapter VI also E. Pardoux [190J. Related results can be found in J. Picard [193J. More recently time reversal has been used to extend Ito's formula for Brownian motion to functions which are more general than just being convex, although we did not present these results in this book. The interested reader can consult for example H. Follmer, P. Protter and A.N. Shiryaev [77] and also H. Follmer and P. Protter [76] for the multidimensional case. The case for diffusions is treated by X. Bardina and M. Jolis [6]. Exercises for Chapter VI Exercise 1. Let (0., F, IF, B, P) be a standard Brownian motion. Expand IF by the initial addition of 0" { B t}. Let M be the local martingale in the formula l tM B1 - B sB t = Bo + Mt + 1 ds.o - s Show that the Brownian motion M = (Mt )O Exercises for Chapter VI 385 Exercises 6 through 9 are linked, with the climax being Exercise 9. Exercise 6. Let B denote standard Brownian motion on its canonical path space on continuous functions with domain JR+ and range JR, with Bt (w) = w(t). IF is the usual minimal filtration completed (and thus rendered right continuous a fortiori). Define ~u = P{B a E UIFt }. Show that ~U = i ~udu where and g(utx)- 1 exp{-(u-x)2} " - V21l"(a-t) 2(a-t) for t < a. Show also that ~u = l{B a Eu} for t ~ a. Exercise 7. (Continuation of Exercise 6.) Show that Yau_ = a except on the null set {Ba = u} and infer that (~U )O::;t 386 Exercises for Chapter VI a-t t it 1Bt - --Bo - -u = (a - t) --d(3s a a 0 a-s under Pu , and thus (a - t) J~(a - s)-ld(3s, which is defined only on [0, a), has the same distribution under Pu as the Brownian bridge, whence i t 1lim (a - t) --d(3s = at--+a 0 a - s a.s., and deduce the desired result.) Exercise 10 (expansion by a natural exponential random variable). Let IF be a filtration satisfying the usual hypotheses, and let T be a totally inaccessible stopping time, with P(T < 00) = 1. Let A = (Atk~o be the compensator of l{t~T} and let M be the martingale M t = l{t~T} - At. Note that AT = Ax). Let G be the filtration obtained by initial expansion of IF with O"{ACX)}. For a bounded, non-random, Borel measurable function f let M! = J~ f(As)dMs which becomes in G, M! = f(AT )l{t~T} - I t f(As)dA s = f(AT )l{t~T} - F(At), where F(t) = J~ f(s)ds, since A is continuous. Show that E{f(At )} E{F(At )}, and since f is arbitrary deduce that the distribution ofthe random variable AT is exponential of parameter 1. Exercise 11. Let IF, T, A, and G be as in Exercise 10. Show that if X is an IF martingale with X T - = X T a.s., then X is also a G martingale. Exercise 12. Show that every stopping time for a filtration satisfying the usual hypotheses is the end of a predictable set, and is thus an honest time. Exercise 13. Let B = (Bt)t~o be a standard three dimensional Brownian motion. Let L = sup{t : II B t II -s:; I}, the last exit time from the unit ball. Show that L is an honest time. Exercise 14. Let M L be the fundamental L martingale of a progressive ex- pansion of the underlying filtration IF using the non-negative random variable L. (a) Show that for any IF square integrable martingale X we have EXL = EXCX)M/;,. (b) Show that M L is the only IFL square integrable martingale with this property for all IF square integrable martingales. Exercise 15. 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Une remarque sur une meme integrale stochastique calculee dans deux filtrations. In Seminaire de Probabilites XVIII, volume 1059 of Lecture Notes in Mathematics, pages 172-178. Springer-Verlag, Berlin, 1984. Symbol Index A L 371 V n 220,301 IIAllp 245 dx(H, J) 155 Jo oo IdA.1 101,245 £(X) 85 A.L ,Ax 180 £R(Z) 320 A 154 £H(Z) 322 A 118 £F-S(Z) 280 IAI, (IAlt)(~o 40, 101 (~ . j)(t,w,x) 279 jAloo 101 (Ftk".o 355 E{IAloo} < 00 111 F·A 40 E{IAITn } < 00 111 IF 3,356 blL 56, 155 IF L 370 bO 102 IF/-< 36 bP 102, 154 FT 5 BMO 195 :Pi 16 Di 252 FT- 105 ]])) 56,244 F; 293,309 ]]))n 250,301 JF 377 ]]))ucp 57 F-S acceptable 279 404 Symbol Index F-S integral 82 FV 39,101 9t 228 (fi2, X)integrable 163 H·A 40 H . X 58,60, 156 HQ . X 60,165,170 He;· X 61 Hg· X 175 [H, Y] 378 HP 244 J HsdXs 58 Jooo HsdXs 58,60 J; H s 0 dXs 82,271 J; HsdXs 60,156 J;+HsdXs 60, 78 lHl( L) 362 I 357lHl fi2 154,244,193 fi2(p) 169 fiP 193 Hoo 314 (HI) 302,328 (H2) 302,328 (H3) 329 Ix 52, 144 I(M) 181,185 Jx 58 jp(N,A) 245 L(X) 163 La(X)t 212,230 Lf 212 L 56,101,102,153,155,244 L(G) 61 L ucp 57 L O 52,56 (M, N) 178 M L 372 M T - 172 M 2 178,186 M 2 (A) 182 IIMIIBMO 195 N 154 o 102 OL 372 pi-' 36 P 102, 154,206 P 293,309 Q rv P 131 Qt(w,dx) 362 Qi 4,214 JRn 19 S 51 S(IH!) 53 Su 52,56 SUCP 57 S(a) 248 S(A) 179 [T] 104 ucp 57 Var(X) 116 Varr(X) 116 X(an ) 269 X* 11,244 X+* 373 X+* 373t Xc 70,221 X T 10,178 X T- 54,172 XTn 369 Xl 293,301 [X, X] 66 [X, X]C 70,70,216,271 [X,Y] 66 [XC,XCj 70 ~Xt 25,60, 158 IIXII,g;p 244 IIXII?i2 244 (X,X) 70, 122 (XC,XC) 70 {X_ :s; a} 214 °xn 369 xVy 211 xAy 211 x+ 211 ya 64,267 ya+ 267 Y- 64,159 yt 377 [Z,Z] 84 Zf 221 (Z : H)t 319 J;(dZs)Hs 319 n 293,309 a-slice 248 J~ 305 ¢ 301,382 ep 311 1rn 17 a(Lr,LOO ) 150 ((x,w) 303 « 131 Symbol Index 405 Subject Index a-sliceable 248 BMO see bounded mean oscillation IF optional projection 369 IF special, G semimartingale 369 F-S see Fisk-Stratonovich FV see finite variation process re norm 154 rfP norm 193 HP (pre)locally in 247 converge (pre)locally in 260 norm 245 fi..P (pre)locally in 247 converge (pre)locally in 260 norm 244 ucp see uniform convergence on compacts in probability absolutely continuous 131 Absolutely Continuous Compensators Theorem 192 absolutely continuous space 191 accessible stopping time 103 adapted process decomposable 55 definition of 3 space lL of with caglad paths 155 with bounded caglad paths 56 with cadlag paths 56 with caglad paths 56 angle bracket 123 announcing sequence for stopping time 103 approximation of the compensator by Laplacians 150 arcsine law 228 associated jump process 27 asymptotic martingale (AMART) 218 autonomous 250,293 Azema's martingale 204 and last exit from zero 228 definition of 228 local time of 230 Backwards Convergence Theorem 9 Banach-Steinhaus Theorem 43 Barlow's Theorem 240 Bichteler-Dellacherie Theorem 144 Bouleau-Yor formula 226,227 bounded functional Lipschitz 257 bounded jumps 25 bounded mean oscillation BMO norm 195 B M 0 space of martingales 195 Duality Theorem 199 bracket process 66 Brownian bridge 97,299,384 Brownian motion absolute value of 217 arcsine law for last exit from zero 228 Azema's martingale 228 Brownian bridge see Brownian bridge completed natural filtration right continuous 19 408 Subject Index conditional quadratic variation of 123 continuous modification 17 continuous paths 221 definition of 16 Fisk-Stratonovich exponential of 280 Fisk-Stratonovich integral for 288 geometric 86 Girsanov-Meyer Theorem and 141 Ito integral for 78 Levy's characterization of 88 Levy's Theorem (martingale characterization) 86,87 last exit from zero 228 last exit time 46 local time of 217, 225 martingale characterization (Levy's Theorem) 86,87 martingale representation for 187 martingales as time change of 88, 187 maximum process 23 on the sphere 281 Ornstein-Uhlenbeck 298 quadratic variation of 17,67, 123 reflection principle 23, 228 reversibility for integrals 380 semimartingale 55 standard 17 starting at x 17 stochastic area formula 89 stochastic exponential for see geometric Brownian motion stochastic integral exists 173 strong Markov property for 23 Tanaka's formula 217 tied down or pinned see Brownian bridge time change of 88 unbounded variation 19 white noise 141,243 Burkholder's inequality 222 Burkholder-Davis-Gundy inequalities 193 cadlag 4 caglM 4 canonical decomposition 129,154 censored data 121 centering function 32 change of time 88, 190 exercises in Chap. II 98 Lebesgue's formula 190 change of variables see Ito's formula, see Meyer-Ito formula Change of Variables Theorem for continuous FV processes 41 for right continuous FV processes 78 classical semimartingale 102,127,144 Class D 106 closed martingale 8 Comparison Theorem 324 compensated Poisson process 31,42, 65 compensator 118 Absolutely Continuous Compensators Theorem 192 Knight's compensator calculation method 151 compensator of l{t;:::L} for IF 371 compound Poisson process 33 conditional quadratic variation 70, 122 Brownian motion 123 polarization identity 123 continuous local martingale part of semimartingale 70,221 continuous martingale part 191 converge (pre)locally in HP, s..p 260 convex functions -- and Meyer-Ito formula 214 of semimartingale 210 countably-valued random variables corollary 365 counting process 12 explosion time 13 without explosions 13 crude hazard rate 122 decomposable adapted process 55, 101 diffeomorphism of jRn 319 diffusion 243,291,297 examples 298 rotation invariant 282 diffusion coefficient 298 discrete Laplacian approximations 150 355,369 148,189 Doleans-Dade exponential see stochastic exponential Dominated Convergence Theorem for stochastic integrals in H r 267 in ucp 174 Doob class see Class D Doob decomposition 105 Doob's maximal quadratic inequality 11 Doob's Optional Sampling Theorem 9 Doob-Meyer Decomposition Theorem case without Class D 115 general case 111 totally inaccessible jumps 106 drift coefficient 143,298 removal of 137 Duality Theorem 199 Dynkin's expectation formula 350 Dynkin's formula 56,350 Einstein convention 305 Emery's example of stochastic integral behaving badly 176 Emery's inequality 246 Emery's structure equation see structure equation Emery-Perkins Theorem 240 enveloping sequence for stopping time 103 equivalent probability measUres 131 Euler method of approximation 353 evanescent 158 example Emery's example of stochastic integral behaving badly 176 expansion via end of SDE 367 Gaussian expansions 366 hazard rates and censored data 121 Ito's example 366 reversibility for Brownian integrals 380 reversibility for Levy process integrals 380 example of local martingale that is not martingale 37,74 example of process that is not semimartingale 217 Subject Index 409 example of stochastic integral that is not local martingale 176 examples of diffusions 298 Existence of Solutions of Structure Equation Theorem 201 expanded filtration 370 expansion by a natural exponential LV. 386 expansion via end of SDE 367 explosion time 13,254,303 extended Gronwall's inequality 352 extremal point 183 Fefferman's inequality 195 strengthened 195 Feller process 35 filtration countably-valued random variables corollary 365 ' definition of 3 expansion by a natural exponential r.v. 386 filtration shrinkage 367 Filtration Shrinkage Theorem 369 Gaussian expansions 366 independence corollary 365 Ito's example 366 natural 16 progressive expansion quasi left continuous right continuous 3 filtration shrinkage 367 Filtration Shrinkage Theorem 369 finite quadratic variation 271 finite variation process definition of 39, 101 integrable variation 111 Fisk-Stratonovich acceptable 278 approximation as limit of sums 284 integral 82,271 integral for Brownian motion 288 Integration by Parts Theorem 278 Ito's circle 82 Ito's formula 277 stochastic exponential 280 flow of SDE 301 of solution of SDE 382 410 Subject Index strongly injective 311 weakly injective 311 Fubini's Theorem for stochastic integrals 207,208 function space 301 functional Lipschitz 250 fundamental L martingale 372 fundamental sequence 37 Fundamental Theorem of Local Martingales 125 Gamma process 33 Gaussian expansions 366 generalized Ito's formula 271 generator of stable subspace 179 geometric Brownian motion 86 Girsanov-Meyer Theorem 132 predictable version 133 graph of stopping time 104 Gronwall's inequality 342,352 extended 352 Hadamard's Theorem 330 Hahn-Banach Theorem 199 hazard rates 121 hitting time 4 Holder continuous 99 honest random variable 373 Hunt process 36 Hypothesis A 221 increasing process 39 independence corollary 365 independent increments of Levy process 20 of Poisson process 13 index of stable law 34 indistinguishable 3,60 infinitesimal generator 349 injective flow see strongly injective flow integrable variation process 111 Integration by Parts Theorem for Fisk-Stratonovich integrals 278 for semimartingales 68,83 intrinsic Levy process 20 Ito integral see stochastic integral Ito's circle 82 Ito's formula for an n-tuple of semimartingales 81 for complex semimartingales 83,84 for continuous semimartingales 81 for Fisk-Stratonovich integrals 277 for semimartingales 78 generalized 271 ItO-Meyer formula see Meyer-Ito formula Ito's example 366 Ito's Theorem for Levy processes 356 Jacod's Countable Expansion 366 Jacod's Countable Expansion Theorem 53,356 Jacod's criterion 363 J acod-Yor Theorem on martingale representation 199 Jensen's Inequality 11 Jeulin's Lemma 360 Kazamaki's criterion 139 Knight's compensator calculation method 151 Kolmogorov's continuity criterion 220 Kolmogorov's Lemma 218 Kronecker's delta 305 Kunita-Watanabe inequality 69 Levy Decomposition Theorem 31 Levy measure 26 Levy process associated jump process 27 bounded jumps 25 cadlag version 25 centering function 32 definition of 20 has finite moments of all orders 25 intrinsic 20 is a semimartingale 55 Ito's Theorem for 356 Levy measure 26 reflection principle 49 reversibility for integrals 380 strong Markov property for 23 symmetric 49 Levy's arcsine law 228 Levy's stochastic area formula 89 Levy's stochastic area process 89 Levy's Theorem characterizing Brownian motion 86,87 Levy-Khintchine formula 31 last exit from zero 228 Le Jan's Theorem 124 Lebesgue's change of time formula 190 Lenglart's Inclusion Theorem 177 Lenglart-Girsanov Theorem 135 linkage operators 329 Lipschitz bounded functional 257 definition of 250,293 functional 250 locally 251,254,303 process 250,311 random 250 local behavior of stochastic integral 62,165,170 local behavior of the stochastic integral at random times 375 local convergence in HP, s.p 260 local martingale -- BMO 195 cadlag martingale is 37 compensator of 118 condition to be a martingale 38, 73 continuous part of semimartingale 221 decomposable 126 definition of 37 example that is not a martingale 37,74 fundamental sequence for 37 Fundamental Theorem 125 intervals of constancy 71, 75 not locally square integrable 126 not preserved under shrinkage of filtration 126 pre-stopped 172 preserved by stochastic integration see stochastic integral, preserves reducing stopping time for 37 time change of Brownian motion 88 local property 38, 162 local time and Bouleau-Yor formula 227 and change of variables formula see Meyer-Ito formula, see Meyer-Tanaka formula and delta functions 217 continuous in t 213 Subject Index 411 definition of Lf 212 discontinuous, example of 225 occupation time density 216, 225 of Azema's martingale 230 of Brownian motion 217 of semimartingale 212 regularity in space 224 support of 214 locally bounded process 164 locally in HP, s.p 247 locally integrable process 359 locally integrable variation process 111 locally Lipschitz 251,254,303 locally special semimartingale 149 Memin's criterion for exponential martingales 352 Metivier-Pellaumail inequality 352 Metivier-Pellaumail method 352 Markov process definition of 34 diffusion 243,291,297 Dynkin's expectation formula 350 Dynkin's formula 56 equivalence of definitions 34 infinitesimal generator 349 simple 291 strong 292 time homogeneous 35, 292 transition function 35, 292 transition semigroup 292 martingale L 2 projection 9 rtP norm 193 BMO 195 asymptotic (AMART) 218 Azema's 204, 228 Backwards Convergence Theorem 9 Burkholder's inequality 222 cadlag modification 8 closed 8 continuous martingale part 191 definition of 7 Doob's inequalities 11 fundamental L martingale 372 in L 2 178 Jacod-Yor Theorem 199 local 37 412 Subject Index Martingale Convergence Theorem 8 martingale representation 203 maximal quadratic inequality 11 measures M 2 (A) 182 not locally square integrable 126 Optional Sampling Theorem 9 orthogonal 179 predictable representation property 182 purely discontinuous part 191 quasimartingale 116 sigma martingale 233 space M 2 of 178 square integrable 73, 74 strongly orthogonal 179 submartingale 7 supermartingale 7 uniformly integrable 8 with exactly one jump 124 Martingale Convergence Theorem 8 martingale representation 203 mathematical finance theory 137 maximal quadratic inequality 11 Meyer's Theorem 104 Meyer-Ito formula 214 Meyer-Tanaka formula 216 modification 3 Monotone Class Theorem 7 monotone vector space 7 closed under uniform convergence 7 multiplicative collection of functions 7 natural filtration 16 natural process 111 net hazard rate 122 Novikov's criterion 140 oblique bracket 123 occupation time density 216, 225 optional a-algebra 102,372 optional projection 367,369 Optional Sampling Theorem 9 Ornstein-Uhlenbeck process 298 orthogonal martingales 179 Ouknine's formula 240 partition 116 path space 142 path-by-path continuous part 70 Perkins-Emery Theorem 240 Picard iteration 255 pinned Brownian motion 299 Pitman's Theorem 240 Poisson process arrival rate 15 compensated 31,42,65,118 compound 33 definition of 13 independent increments 13 intensity 15 stationary increments 13 polarization identity 66, 123, 271 potential 105, 150 pre-stopped local martingale 172 predictable a-algebra 102, 154 predictable compensator of l{t;:::L} for IF 371 predictable process predictable a-algebra 102 simple 51 predictable projection 368 predictable representation property 182 predictable stopping time 103 predictably measurable process 102, 105 prelocal convergence in HP, {iP 260 prelocally in HP, {iP 247 - preserved by stochastic integration see stochastic integral, preserves process Lipschitz 250,311 process stopped at T 10, 37 progressive expansion 355, 369 projections optional projection 367 predictable projection 368 property holds locally 38,162,247 property holds prelocally 162,247 purely discontinuous part of martingale 191 quadratic covariation 66,378 polarization identity 66,271 quadratic covariation process 270 quadratic pure jump 71 quadratic variation conditional 122 continuous part of 70, 271 covariation 66 finite 271 of a semimartingale 66 polarization identity 66, 271 quasi left continuous filtration 148, 189 quasimartingale 116 random Lipschitz 250 random partition definition of 64 tending to the identity 64 Rao's Theorem 118 Ray-Knight Theorem 225 reduced by stopping time 37 reflection principle for Brownian motion 23,228 for Levy processes 49 for stochastic area process 92 regular conditional distribution 362 regular supermartingale 150 representation property 182 reversibility for Brownian integrals 380 reversibility for Levy process integrals 380 reversible semimartingale 378 Riemann-Stieltjes integral 41 right continuous filtration 3 right stochastic exponential 319,320 right stochastic integral 319 rotation invariant diffusion 282 sample paths of stochastic process 4 scaling property 238 semimartingale (IF, G) reversible 378 ere, X) integrable 163 re norm 154 absolute value of see Meyer-Tanaka formula bracket process 66 canonical decomposition 129 classical 102,127,144 continuous local martingale part 70, 221 convex function of 210 definition of 52 equivalent norm 244 Subject Index 413 example of process that is not 217 Integration by Parts Theorem 68, 83 local martingale as 127 local time of 212 locally special 149 preserved by stochastic integration see stochastic integral, preserves quadratic pure jump 71 quadratic variation of 66 quasimartingale as 127 space 1t2 of 154 special 129, 154 stochastic exponential of 85 submartingale as 127 supermartingale as 127 topology 264 total 52 semimartingale topology 264 separable measurable space 189 sharp bracket 123 sigma martingale 233 preserved by stochastic integration see stochastic integral, preserves sign function 212 signal detection 140 simple Markov process 291 simple predictable process 51 Skorohod topology 220 Skorohod's Lemma 239 smallest filtration making LV. a stopping time 119,370 special semimartingale 129,154 canonical decomposition 129 square integrable martingale 178 preserved by stochastic integration see stochastic integral, preserves stable law 34 index 34 stable process defintion of 33 scaling properties 34 symmetric 34 stable subspace generated 179 of M 2 178 stable under stopping 178 standard Borel space 362 stationary Gaussian process 299 414 Subject Index stationary increments of Levy process 20 of Poisson process 13 statistical communication theory 140 Stieltjes integral see Riemann-Stieltjes integral stochastic area formula 89 stochastic area process definition of 89 density 91 Levy's formula 89 properties 92 reflection principle 92 stochastic differential equation 255, 321 flow of 301 flow of solution 382 weak solution 201,240 weak uniqueness 201 stochastic exponential appproximation of 269 as a diffusion 298 definition of 85 Fisk-Stratonovich 280 for Brownian motion see geometric Brownian motion for semimartingale 85 invertibility of 335 right 319,320 uniqueness of 255 stochastic integral X integrable, L(X) 163 Associativity Theorem 62, 159, 165 behavior of jumps 60, 158, 165 does not preserve FV processes in general 241 Dominated Convergence Theorem in H r 267 Dominated Convergence Theorem in ucp 174 example that is not local martingale 176 for lL 59 for bP 156 for S 58 for P 163 for Brownian motion 78 Fubini's Theorem 207,208 local behavior at random times 375 local behavior of 62, 165, 170 preserves FV processes, integrands in lL 63 continuous local martingales 173 local martingales 128,171,174 locally square integrable local martingales 63, 171 semimartingales 63 sigma martingales 234 square integrable martingales 159, 171 right 319 said to exist 163 stochastic process adapted 3 decomposable 55,101 definition of 3 finite variation 39,101 increasing 39 indistinguishable 3, 60 locally bounded 164 locally integrable 359 modification of 3 natural 111 potential 105 sample paths of 4 stopped 10,37 strong Markov 36 stopped process 10,37 stopping time a-algebra 5, 105 accessible 103 announcing sequence 103 change of time 98 definition of 3 enveloping sequence 103 explosion time 13,254,303 fundamental sequence of 37 graph 104 hitting time 4 hitting time of Borel set is 5 predictable 103 reducing 37 totally inaccessible 103 Stratonovich integral see Fisk- Stratonovich, integral Stratonovich Integration by Parts Theorem 278 strengthened Fefferman inequality 195 Stricker's Theorem 53 strong Markov process 36,292 strong Markov property 36,292 for Brownian motion 23 for Levy process 23 strongly injective flow 311 strongly orthogonal martingales 179 structure equation 200, 238 Existence of Solutions Theorem 201 submartingale 7 supermartingale 7 regular 150 symmetric Levy process 49 symmetric stable process 34 Tanaka's formula 217 tied down Brownian motion 299 time change 190 time change of Brownian motion 88 time homogeneous Markov process 292 time reversal 377 Subject Index 415 total semimartingale 52 total variation process 40 totally inaccessible stopping time 103 transition function for Markov process 292 transition semigroup 292 truncation operators 265 uniform convergence on compacts in probability 57 uniformly integrable martingale 8 usual hypotheses 3, 291 variation 116 variation along T 116 weak solution of SDE 201, 240 weak uniqueness of SDE 201 weakly injective flow 311 white noise 140, 243 Wiener measure 141 Wiener process 137, 140 Applications of Mathematics (continued from page II) 51 Asmussen, Applied Probability and Queues (2nd ell. 2003) 52 Robert, Stochastic Networks and Queues (2003) 53 Glasserman, Monte Carlo Methods in Financial Engineering (2003)