Stochastic Integration and Differential Equations

Stochastic Mechanics Random Media Signal Processing and Image Synthesis Mathematical Econ omics and Finance Stochastic Optimization

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Applications of Mathematics 1 Fleming/Rishel, Deterministic and Stochastic Optimal Control (1975)

2 Marchuk, Methods of Numerical 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 of Random Processes I: General Theory (1977, znd. 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) u

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 17 18 19 20 21

Ibragimov/Has'minskii, Statistical Estimation: Asymptotic Theory (1981) Cesari, Optimization: Theory and Applications (1982) Elliott, Stochastic Calculus and Applications (1982) Marchuk/Shaidourov, Difference Methods and Their Extrapolations (1983) Hijab, Stabilization of Control Systems (1986) Protter, Stochastic Integration and Differential Equations (1990, znd. 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 Kushner/Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time (1992) 25 Fleming/Soner, Controlled Markov Processes and Viscosity Solutions (1993) 26 Baccelli/Bremaud, Elements of Queueing Theory (1994, 2nd ed. 2003) 27 Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods (1995, 2nd. ed. 2003) 28 Kalpazidou, Cycle Representations of Markov Processes (1995) 29 Elliott/Aggoun/Moore, Hidden Markov Models: Estimation and Control (1995) 30 Hermlndez-Lerma/Lasserre, Discrete-Time Markov Control Processes (1995) 31 Devroye/Gyorfi/Lugosi, 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 Duflo, Random Iterative Models (1997) 35 Kushner/Yin, Stochastic Approximation Algorithms and Applications (1997) 36 Musiela/Rutkowski, Mar tingale Methods in Financial Modelling (1997) 37 Y in, Continuous-Time Markov Chains and Applications (1998) 38 Dembo/Zeitouni, Large Deviations Techniques and Applications (1998) 39 Karatzas, Methods of Mathematical Finance (1998) 40 Fayolle/Iasnogorodski/Malyshev, 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) so 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 U SA e-mail: [email protected]

Managing Editors B. Rozovskii

M. Yor

Center for Applied Mathematical Sciences University of Southern California 1042 West 36 th Place, Denney Research Building 308 Los Angeles, CA 90089, USA

Universite de Paris VI Laboratoire de Probabilites et Modeles Aleatoires 175, rue du Chevaleret 75013 Paris, France

Mathematics Subject Classification (2ooo): PRIMARY: 60H05, 60H10, 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.

Bibliographic 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 41/3142DB-543210

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. I. 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'-TEX, 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

.................................................. 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Basic Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 The Poisson Process and Brownian Motion . . . . . . . . . . . . . . . . 1 2 4 Levy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Why the Usual Hypotheses? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 Local Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7 7 Stieltjes Integration and Change of Variables . . . . . . . . . . . . . . . 39 8 Naive Stochastic Integration Is Impossible . . . . . . . . . . . . . . . . . . 43 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Exercises for Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 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 Semimartingales and Decomposable Processes . . . . . . . . . . . . . 101 101 1 Introduction 2 The Classification of Stopping Times 103 3 The Doob-Meyer Decompositions . 105 4 Quasimartingales . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 1 16

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Contents

XII

Compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 The Fundamental Theorem of Local Martingales . . . . . . . . . . . . 12 4 Classical Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Girsanov's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 The Bichteler-Dellacherie Theorem . . . . . . . . . . . . . . . . . . . . 143 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Exercises for Chapter III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7

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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 . . . . . . . . . . . . . . . . . . . 2 00 6 Stochastic Integration Depending on a Parameter . . . . . . . . . . . 2 05 7 Local Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 10 8 Azema' s Martingale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7 9 Sigma Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 33 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 35 Exercises for Chapter IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 36 .

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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 . . . . . . . . . . . . . . . 25 7 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 . . . . . . . 32 1 10 Flows as Diffeomorphisms: The General Case . . . . . . . . . . . . . . . 328 1 1 Eclectic Useful Results on Stochastic Differential Equations . . . 338 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Exercises for Chapter V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 .

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VI Expansion of Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 2 Initial Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 3 Progressive Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 4 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 77 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Exercises for Chapter VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 .

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XIII

Contents

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389 403 407

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. Rao'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 1i1 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)ot :Fu, all t, 0 < t < oo; that is, the filtration IF is right continuous. We always assume that the usual hypotheses hold.

Definition. A random variable T 0 � [0, oo] is a stopping time if the event {T < t} E :Ft, every t, 0 < t < oo. One important consequence of the right continuity of the filtration is the following theorem. Theorem 1. The event {T < t} E :Ft, 0 < t < oo, if and only if T is a :

stopping time. Proof Since {T < t} = nt +ou>t {T < u}, any > 0, we have {T < t} E nu>t :Fu = :Ft , so T is a stopping time. For the converse, {T < t} = D Ut> e > o {T < t - } and {T < t - c:} E Ft -e:, hence also in :Ft. A stochastic process X on (0, :F, P) is a collection of JR.-valued or JR.d_ valued random variables (Xt)o
c:

,

about the concept of equality of two stochastic processes. Definition. Two stochastic processes X and Y are modifications if Xt = yt a.s. , each t. Two processes X and Y are indistinguishable if a.s., for all t,

Xt = yt.

4

I Preliminaries

If X and Y are modifications there exists a null set, Nt, such that ifw � Nt, then Xt(w) = yt(w). The null set Nt depends on t. Since the interval [O, oo ) is uncountable the set N = Uo
Definition. A stochastic process X is said to be dtdlag if it a.s. has sam­

ple paths which are right continuous, with left limits. Similarly, a stochastic process X is said to be caglad if it a.s. has sample paths which are left continuous, with right limits. (The nonsensical words cadtag and caglad are acronyms from the French for continu a droite, limites a gauche and continu a gauche, limites a droite, respectively.)

Let X and Y be two stochastic processes, with X a modifica­ tion of Y. If X and Y have right continuous paths a.s., then X and Y are indistinguishable. Proof Let A be the null set where the paths of X are not right continuous , and let B be the analogous set for Y. Let Nt = {w : Xt(w) -1- Yt(w)}, and Theorem 2.

let N = UtEQNt, where Q denotes the rationals in [0, oo) . Then P(N) = 0. Let M = AU BUN, and P(M) = 0. We have Xt(w) = yt(w) for all t E Q, w � M. If t is not rational, let tn decrease to t through Q. For w � M , Xtn (w) = Ytn (w), each n, and Xt(w) = lim n->oo Xtn (w) = limn -.00 YtJw) = D yt(w). Since P(M) = 0, X and Y are indistinguishable.

Corollary. Let X and Y be two stochastic processes which are cadlag. If X

is a modification of Y, then X and

Y are indistinguishable.

Cadlftg processes provide natural examples of stopping times.

Definition. Let X be a stochastic process and let A be a Borel set in Define

T(w) = inf{t > 0: Xt

R

E A}.

Then T is called a hitting time of A for X.

Let X be an adapted cadlag stochastic process, and let A be an open set. Then the hitting time ofA is a stopping time. Proof By Theorem 1 it suffices to show that {T < t} E :Ft , 0 < t < oo. But

Theorem 3.

{T

<

t} = U

sEQ n[O,t)

{ Xs E A } ,

since A is open and X has right continuous paths. Since { Xs :F8, the result follows.

E A}

= X;1(A) E

D

1

Basic Definitions and Notation

5

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}

Theorem 4.

is a stopping time. Proof By Xt_(w) we mean lim8_,t,s
set and

0

n sEQn[O,t)

It is a very deep result that the hitting time of a Borel set is a stopping time. We do not have need of this result. The next theorem collects elementary facts about stopping times; we leave the proof to the reader.

Theorem 5.

Let S,

T

times: (i) S 1\ T = min(S, T); (ii) S V T max(S, T); (iii) S + T; (iv) aS, where a > 1.

be stopping times. Then the following are stopping

=

The O"-algebra :Ft 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 :Fr is defined to be

{A

E

:F: An {T < t}

E

:Ft, all t > 0}.

The previous definition is not especially intuitive. However it does well represent "knowledge" up to time T, as the next theorem illustrates.

Let T be a finite stopping time. Then :Fr is the smallest algebra containing all cadlag processes sampled at T. That is, :Fr O"{ Xr; X all adapted cadlag processes}. Proof Let g = O"{ Xr; X all adapted dtdlag processes } . Let A E Fr. Then

Theorem 6.

O"­

=

Xt = lAl{t>r} 1 is a cadlag process, and Xr = lA. Hence A 1

1A

is the indicator function of A

:

1A (w)

=

{

1

'

0,

wE A, w �A.

E

Q, and :Fr

c

Q.

6

I Preliminaries

Next let X be an adapted dtdlag process. We need to show Xr is Fr measurable. Consider X(s, w) as a function from [0, oo) x 0 into R Construct cp : { T < t} � [0, oo) x 0 by cp(w) = (T(w), w) . Then since X is adapted and cadlag, we have Xr = X ocp is a measurable mapping from ( {T < t}, Ftn{T < t}) into (IR, !3) , where l3 are the Borel sets of R Therefore {w: X(T(w),w) E

is in Ft, and this implies Xr

E

B} n {T < t}

Fr . Therefore Q C Fr.

D

We leave it to the reader to check that if S < T a.s., then Fs C Fr, and the less obvious (and less important) fact that Fs nFr = Fs11r· If X andY are cadlag, then Xt = yt a.s. each t implies that X andY are indistinguishable, as we have already noted. Since fixed times are stopping times, obviously if Xr = Yr a.s. for each finite stopping time T, then X and Y are indistinguishable. If X is cadlag, let .6.X denote the process .6.Xt = Xt - Xt - . Then .6.X is not cadlag, though it is adapted and for a.a. w , t � .6.Xt = 0 except for at most countably many t. We record here a useful result. 7.

Let X be adapted and cadlag. If .6.Xr1 { r 0} is countable a.s. since X is cadlag. Moreover Theorem

:

1 {t : I .6.Xtl > 0} = u {t : 1 .6.Xt l > -} n n= 1 and the set { t I.6.Xt l > 1 / } must be finite for each n, since to < oo. Using Theorem 4 we define stopping times for each n inductively as follows: Tn ' 1 = . f{ t > 0: I.6.Xtl > -n1 } 1 k n , k > n r , inf{ t > T 1 I.6.Xt l n }. Then Tn , k > Tn k 1 a.s. on { Tn , k 1 < oo }. Moreover, 00

n

:

m

=

,

-

-

:

-

{f.6.Xt l > 0} = U {f .6.Xrn , d{rn,k 0}, n ,k where the right side of the equality is a countable union. The result follows. Corollary. Let T, .6.Xr 1 { r
D

X and Y be adapted and dtdlag. If for each stopping time .6.Yr 1 { r
=

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 [41] or Dellacherie-Meyer [45]. 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 n satisfying the three conditions: (i) 1i is a vector space over lR; (ii) 1n E 1i (i.e., constant functions are in H); and < fn < , and limn oo fn = f , (iii) if Un ) n >l C H, and 0 < ft < h < 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, g E M implies that fg E M. ·

·

·

·

·

·

_,

For a collection of real-valued functions M defined on n , 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 {f - 1 (A); A E .B(lR), f E M}. 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 1i is a monotone vector space containing M, then 1i contains all bounded, A measurable functions. Theorem 8 is proved in Dellacherie-Meyer [45, page 14] with the additional

hypothesis that 1i 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 365].) 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
Definition. A real-valued, adapted process X = (Xt)o
E{XtiFs } < X8 , resp. > X8).

8

I

Preliminaries

Note that martingales are only defined on [0, oo); that is, for finite t and not t = oo. It is often possible to extend the definition to t = oo.

Definition. A martingale X is said to be closed by a random variable Y if E{JYI} < oo and Xt = E{YIFt}, o < t < oo. 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.

Let X be a supermartingale. The function t E{Xt } is right continuous if and only if there exists a modification Y of X which is cadlag. Such a modification is unique. f-->

Theorem 9.

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{Xt} is constant, and hence it has a right continuous modification. Corollary. If X = (Xt)o < t< oo is a martingale then there exists a unique modification Y of X which is dtdlag. Since all martingales have right continuous modifications, we will always assume that we are taking the right continuous version, without any special mention. Note that it follows from this corollary and Theorem 2 that a right continuous martingale is cadlag. Theorem 10 (Martingale Convergence Theorem) . Let X be a right continuous supermartingale, supo< t oo }{IUal>n} a

Let (Ua )aEA be a subset of L1. The following are equivalent: {i) (Ua)aEA is uniformly integrable. {ii) supaEA E{J Ua l } < oo, and for every c > 0 there exists 8 > 0 such that A E F, P(A) < 8, imply E{IUalA I } < c. 2 Vo:o;t
Theorem 11.

<

2

Martingales

9

(iii) There exists a positive, increasing, convex function G(x) defined on [O, oo) such that Iim x-+ oo �x) = +oo and supa E{G o IUa l } < oo. 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 Xt a.s. exists, E{I Y I} < oo, 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_. 00 Xt exists a.s., E{ I Y I } < oo, and is a martingale, where Xoo = Y. If X is a uniformly integrable martingale, then Xt converges to X00 = Yin L 1 as well as almost surely. The next theorem we use only once ( in the proof of Theorem 28), but we give it here for completeness. The notation (Xn ) n
->

(Xt )o
A less probabilistic interpretation of martingales uses Hilbert space theory. Let Y E L2(fl, F, P). Since FtC F, the spaces L2(fl,Ft, P) form a family of Hilbert subspaces of L 2 (fl, F, P). Let 11"ty denote the Hilbert space projection of Y onto £2 (0., Ft, P).

Let Y E L 2 (fl, F, P). The process Xt = 11"ty is a uniformly integrable martingale. Proof It suffices to show E{Y!Ft} = 11" t Y. The random variable E{Y!Ft} is the unique Ft measurable r.v. such that JA YdP = JA E{Y!Ft }dP, for any event A E Ft. We have fA YdP = fA 11"tYdP + fA(Y _11", Y)dP. But JA (Y _11"tY)dP = J l A (Y _11"tY)dP. Since lA E L 2 (fl, Ft, P), and (Y - 11"
The next theorem is one of the most useful martingale theorems for our purposes. Theorem 16 (Doob's Optional Sampling Theorem) . Let X be a right

continuous martingale, which is closed by a random variable X00 • Let S and T be two stopping times such that S < T a.s. Then Xs and XT are integrable and Xs = E{XT!Fs } a.s. Theorem 16 has a similar version for supermartingales.

10

I Preliminaries 17.

Let X be a right continuous supermartingale {resp. martin­ gale), and let S and T be two bounded stopping times such that S < T a.s. Then X s and XT are integrable and X s > E{Xr lFs} a.s. (resp. = ) . If Tis a stopping time, then so is t 1\ T min (t, T) , for each t > 0. Definition. Let X be a stochastic process and let Tbe a random time. xr is said to be the process stopped at T if xr Xti\ T · Note that if X is adapted and dtdlag and if T is a stopping time, then Theorem

=

=

is also adapted. A martingale stopped at a stopping time is still a martingale, as the next theorem shows. Theorem 18. Let X be a uniformly integrable right continuous martingale, and let T be a stopping time. Then xr = (XtAT )o
integrable right continuous martingale. Proof xr is clearly right continuous. By Theorem 16 XtAT E{Xrl FtAT } E{Xr 1 {T t} I FtAT } Xr1{T t} I Ft AT } . However for H E Ft we have H1 {T >t} E Fr . Thus, =

=

=

Therefore

Xti\T Xr 1 { Tt} E{XrlFt}, since Xr 1{ T
=

=

=

times. Then

E{E{Y IFs} I Fr} = E{E{Y IFr} IFs} E{YJFsi\T }. =

2 Martingales

Proof

11

yt = E{Y I Ft} · Then yT is a uniformly integrable martingale and YsAT = YJ = E{Yri Fs} = E{E{Y I Fr }IFs}. Interchanging the roles of T and S yields YsAT = Y,J = E{Ys i Fr} = E{E{Y iF s} IFr } . 0 Finally, E{YIFsAT } = YsAT · Let

The next inequality is elementary, but indispensable.

Let cp : lR---. lR be convex, and let X and cp(X) be integrable random variables. For any a-algebra Q, cpoE{XIQ} < E{cp(X) IQ}. Corollary 1. Let X be a martingale, and let cp be convex such that cp(Xt) is integrable, 0 < t < oo. Then cp(X) is a submartingale. In particular, if M is a martingale, then I MI is a submartingale. Corollary 2. Let X be a submartingale and let cp be convex, non-decreasing, and such that cp(Xt)o
We end our review of martingale theory with Doob's inequalities; the most important is when p = 2.

Let X be a positive submartingale. For all p > 1, with q con­ jugate to p {i. e., ; + � = l }, we have Theorem 20.

We let X* denote sup8 IXs l · Note that if M is a martingale with M00 E then IMI is a positive submartingale, and taking p = 2 we have

L2 ,

This last inequality is called Doob's maximal quadratic inequality. An elementary but useful result concerning martingales is the following. Theorem 21. Let X = (Xt)o
Suppose E{IXr l } < oo and E{Xr} = 0 for any stopping time T, finite or not. Then X is a uniformly integrable martingale. Proof Let 0 < s < t < oo, and let A E F8• Let u, if w E A, = if w rt A. oo, UA

{

I Preliminaries

12

Then

UA are stopping times for all u > s. Moreover f

}A

XudP

=

=

J XuAdP- lo.\f A X00dP -

f

lo.\A

XX)dP

E{XuA} 0 by hypothesis, for u > s. Thus for A E F8 and s < t, E{Xt1 A } E{Xs1 A } -E{X00 1n\ A }, which implies E{XtiFs} Xs, and X is a martingale, 0 < t < oo. 0 Definition. A martingale with X0 0 and < oo for each t > 0 is called a square integrable martingale. If E{ < oo as well, then is called an L 2 martingale. Clearly, any L2 martingale is also a square integrable martingale. See also Sect. 3 of Chap. IV. since

=

=

=

=

X

=

E{Xl} X�,}

X

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 (!1, F, IF, P) satisfying the usual hypotheses. Let (Tn)n >O be a strictly increasing sequence of positive random variables. We always takeT0 = 0 a.s. Recall that the indicator function l{t>T n} is defined as if t > Tn(w), if t < Tn(w). Definition. The process N = (Nt)o
=

L 1{t>Tn}

n>l

with values in NU{oo } where N = {0, 1 , 2, . . . } is called the counting process associated to the sequence (Tn)n>l· If we set T

=

supn Tn, then [Tn,oo) = {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

N.

13

N

The random variable T is the explosion time of If T = oo a.s., then is a counting process without explosions. For T = oo, note that for 0 < s < t < oo we have Nt - s = L l {sl The increment Nt counts the number of random times Tn that occur between the fixed times s and t. As we have defined a counting process it is not necessarily adapted to the filtration IF. Indeed, we have the following.

N

- N8

N

counting process is adapted if and only if the associated random variables (Tn ) n >l are stopping times. Proof If the (Tn ) n >O are stopping times (with To = 0 a.s.) , then the event

Theorem 22. A

N

N

for each n. Thus Nt EFt and is adapted. If is adapted, then { Tn < t } { Nt > n} E Ft , each t, and therefore Tn is a stopping time.

=

D

Note that a counting process without explosions has right continuous paths with lef t limits; hence a counting process without explosions is cadlag. Definition. An adapted counting process is a Poisson process if is independent of F8 ; (i) for any s , t, 0 < s < t < oo, (ii) for any s , t, u, v , 0 < s < t < oo, 0 < u < v < oo, t - s = v - u, then the is the same as that of distribution of Nt -

N8

Nt- N8

N

Properties (i) and (ii) are known respectively as the past, and stationary increments. Theorem 23. Let be a Poisson process. Then

Nv- Nu.

increments independent of

N

n = 0, 1, 2, . . . , for some A > 0. That is, Nt has the Poisson distribution with parameter At. Moreover, is continuous in probabilit?} and does not have explosions. Proof The proof of Theorem 23 is standard and is of ten given in more ele­ mentary courses ( cf. , e.g. , Qinlar [33, page 71] ) . We sketch it here. Step 1. For all t > 0, P ( t = 0) = e - >.t , for some constant A > 0.

N

N

3 N is continuous in probability means that for

limit is taken in probability.

t > 0, limu___,t Nu

=

Nt where the

14

I

Preliminaries

Since {Nt = 0 } = {Ns = 0} n {Nt independence of the increments,

- Ns = 0} for 0 < s < t

<

oo

by the

P(Nt = 0) = P(Ns = O)P(Nt - Ns 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 < oo. 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 > 0. If a(t) = 0 it would follow that Nt( w ) = oo a.s. for all t which would con­ tradict that N is a counting process. Note that limu-+t P(INu - Nt l > c) limu _,t P(INu - tl > c) limv_,o P(Nv > c) = Iimv_,o 1 - e - >-v = 0; hence N =

=

=

=

is continuous in probability. Step 2. P(Nt > 2) is o(t). (That is, limt_,o �P(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 ! {3 (t) = 0 is equivalent to showing that limn_,oo n{3( �) = 0. 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 i.i.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+l a.s. Since Sn < N1 , if E { N1 } < oo we can use the Dominated Convergence Theorem to conclude limn ---+ oo nf3 (,� ) = limn -+oo E{Sn } = 0. (That E {N1 } < oo is a consequence of Theorem 34, established in Sect. 4). Also note that E{ N!} < oo implies N1 < oo 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_,00 Sn (w) = 0 a.s. Step 3. limt_,o ! P{ Nt = 1} = A. Since P{Nt = 1 } = 1 - P{Nt = 0} - P{Nt > 2 }, it follows that

t + (t) .hm -P{Nt 1 1 e >. = A. = 1} = hm t-+ 0 t -+0 t t -

Step 4.

o

Conclusion. We write cp(t) = E{a N }, for 0 < a < 1. Then for 0 < s < t < oo, the independence and stationarity of the increments implies that cp( t + s) = cp(t)cp(s) which in turn implies that cp(t) = eti/J(a) . But '

3 The Poisson Process and Brownian Motion

15

cp(t) = L o:n P(Nt n) ()()

=

n=O

()()

= P(Nt 0) + o:P(Nt 1) + L o:n P(Nt = n), n 2 and '1/J(o:) = cp'(O), the derivative of cp at 0. Therefore P(Nt = 0) - 1 o:P(Nt = 1) 1 1.l 1. cp(t) - 1 + + - 0 (t '1/J ( ) = t lm0 t-m-+0 --+ t t t t = -A + Ao: =

=

=

0:

=

{

)}

.

Therefore

cp(t) e =

-

>-

t+>. at , hence

Equating coefficients of the two infinite series yields

0

for n = 0, 1, 2, . . . .

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 < oo, E{Nt} < oo and

E{Nt - Ns iFs } = A(t - s). 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 < oo. The analogous statement holds for (Nt - At)2 - At.

D

16

I Preliminaries

Definition. Let H be a stochastic process. The natural filtration of H, denoted IF0 = (f?)o < t< oo , is defined by f? = a { H8 ; s < t } . That is, f? is the smallest filtration that makes H adapted. Note that natural filtrations are not assumed to contain all the P-null sets of F.

Let N be a counting process. The natural filtration of N is right continuous. Proof Let E [0, oo] and !3 be the Borel sets of E, and let r be the path

Theorem 25.

=

space given by

Define the maps

7rt

:

0 ----+

r

sE [O,oo)

s E [O , oo) by

1rt (w) s =

>--+

Ns/\t(w).

Thus the range of 7rt is contained in the set of functions constant after t. The a-algebra f? is also generated by the single function space-valued random variable 7rt· Let A be an event in nn > l �+ � . Then there exists a set An E ® s E [O, oo) !38 such that A = {nt+ � E An}· Next set Wn = {nt = 1rt + � }. For each w, there exists an n such that s >--+ N8 (w) is constant on [t, t + ;, ] ; therefore n = Un > l Wn, where Wn is an increasing sequence of events. Therefore

A = lim n ( Wn n A) lim n ( Wn n { 1rt+ l E An}) lim n ( Wn n { 7rt E An}) =

n

=

which implies A E f?. We conclude nn >l �+ � equal.

c

f?, which implies they are D

We next turn our attention to the Brownian motion process. Recall that we are assuming as given a filtered probability space (0, F, IF, P) that satisfies the usual hypotheses. Definition. An adapted process B = (Bt)o
( i) for 0 < s < t < oo,

Bt - B8 is independent of F8 (increments are indepen­

dent of the past); (ii) for 0 < s < t, Bt - Bs is a Gaussian random variable with mean zero and variance matrix (t - s)C, for a given, non-random matrix C.

3 The Poisson Process and Brownian Motion

The Brownian motion

starts at x if P(Bo x) =

=

17

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{/Bo/} < oo. 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. of B which has continuous paths a.s.

Then there exists a modification

Theorem 27. Let B = (Bt)o
dimensional standard Brownian martingale.

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 Bo = x for some x E JR.n , is a standard Brownian motion. Note that for an JR.n standard Brownian motion B, writing Bt = (Bl , . . . , BJ:) , 0 < t < oo,i then each Bi is an JR. 1 Brownian motion with continuous paths, and the B 's are independent. We have already observed that a Brownian motion B with E{/Bo/} < oo is a martingale. Another important elementary observation is the following.

Proof. E{Mt} E{Bf - t} 0. Also E{Mt - Ms / Fs } E{B[ - B 'f - (t - s) /Fs}, =

=

=

and

E{BtBs / Fs} BsE{Bt /Fs } B; , since B is a martingale with B8 , Bt E L 2 • Therefore E{Mt - Ms /Fs} E{B[ - 2BtBs + B'f - (t - s) / .Fs} E{(Bt - Bs) 2 - (t - s) / Fs} E{(Bt - Bs) 2 } - (t - s) =

=

=

=

=

=

0, 0

due to the independence of the increments from the past.

Theorem 28. Let 7rn be a sequence of partitions of [a, a+t] . Suppose 7rm C 7rn if m > n (that is, the sequence is a refining sequence). Suppose more­ over that limn ->oo mesh( 7rn ) = 0. Let 7rn B = "':.t E JBt - Bt; )2• Then limn->oo 7rn B = t a.s., for a standard Brownian motion B . ,

7r

i+

I

18

I Preliminaries

Proof.

We first show convergence in mean square. We have 7rnB - t =

2::= {(Bt'+, - Bt; ) 2 - (t i+ 1 - ti ) } t iE 'Trn i

where Yi are independent random variables with zero means. Therefore

E{(7rn B - t) 2 } = E{ ( l::= Yi ) 2 } = 2::= E{Y? }. i i

Next observe that (Bti + l - BtJZ/ (ti+1 - 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 oo. 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) = Ln B = L (Bt; +1 (w) - Bt; (w)) 2 , tiE 'IT-n

for n = -1, -2, -3, . . . . Then it is straightforward (though notationally messy) to show that

E{Nn l Nn

-

1

, Nn - 2 , . . . } = Nn - 1 ·

Therefore Nn is a martingale relative to On = { Nk k < n}, n = -1, -2, . . . By Theorem 14 we deduce limn---+ - oo Nn = limn---+ oo 7rn B exists a.s., and since 1r B converges to t in £2 , we must have limn ---+ oo 1rn B t a.s. as well. 0 n u

,

.

=

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 Ln mesh(7rn ) < oo 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

For almost all the sample paths t Bt ( ) 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 w,

Theorem 29.

�

w

be

where P are all finite partitions of [a, b]. Suppose P(VA < oo) > 0. Let 1r n be a sequence of refining partitions of [a, b] with lim n mesh(7rn ) = 0. Then by Theorem 28 on {VA < oo } , lim b - a = n�oo

=0

""" L.-t

ti E'Tr n

(Bt;+1 - Bt; ) 2

'

since SUPt; E 7rn IBt<+1 - Bt; I tends to 0 a.s. as mesh(7rn ) 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 = oo a.s. Since the null set can depend on the interval [a, b], we only consider intervals with rational endpoints a, b with a < b. Such a collection is countable, and since any interval (a, b) = u� [an , bn ] with a n , 0 bn rational, we can omit the dependence of the null set on the interval. l

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 1Rn -valued processes; for notational simplicity, however, we will consider only IR-valued processes. 4 Once again we recall that we are assuming given a filtered probability space (D, :F, IF, P) satisfying the usual hypotheses. 4

lRn denotes n-dimensional Euclidean space. JR+

real numbers.

=

[0, oo) denotes the non-negative

20

I Preliminaries

Definition. An adapted process process if

X

=

(Xt)t>O

with

Xo

=

0

a.s. is a Levy

X has increments independent of the past; that is, Xt - Xs is independent of :F8 , 0 < s < t < oo; and (ii) X has stationary increments; that is, Xt - Xs has the same distribution as Xt - s , 0 < s < t < oo; and (iii) Xt is continuous in probability; that is, limt---+s Xt Xs , where the limit (i)

=

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 = (Xt)t>o with Xo = 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) Xt is continuous in probability. Of course, an intrinsic Levy process is a Levy process for its minimal (completed) filtration. If we take the Fourier transform of each Xt we get a function f ( t, u ) = ft ( u) given by -

ft(u) E{eiu X } 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'l/!(u)}, for some continuous function '1/J( u ) with '1/J(O) 0. (Bochner's Theorem can be used to show the converse. If '1/J is continuous, '1/J(O) 0, and if for all t > 0, ft(u) e - t'I/J (u ) satisfies 2: i,j ai aj ft(u i - uj ) > 0, for all finite ( u 1 , . . . , un; a 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 p, there exists a Levy process X such that p, 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. Proof. Let M;: J, ( u) . For each fixed u in Q, the rationals in IR, the process (Mtu )oO is a (complex-valued) martingale and thus for a.a. the functions =

iuX t

w

4 Levy Processes

Mtu (w )

and t f---> dtdlag functions. Let

t

f--->

A = {(w ' u) E n

X

e iuX, (w) ,

with

tE

21

IQl+ , are the restrictions to IQl+ of

IR .. e iuX, (w) ' t E /(1\ "'!+ , is not the restriction of a dtdlag function } .

One can check that A is a measurable set. Furthermore, we have seen that J l A (w, u)P(dw) = 0, each u E R By Fubini's Theorem

J 1: l A (w, u)duP(dw) 1: J A (w, u)P(dw)du 1

=

=

0,

hence we conclude that for a.a. w the function t f---> e iu X, (w) , t E Q+ is the restriction of a dtdlag 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) = lim s E IQI s t Xs(w) for all w in the projection onto n of {D x IR} \ A and yt = 0 on A, all t. Since :Ft contains all the P-null sets of :F and (:Ft)O
,

t

The next lemma was used in the proof of Theorem 30. Although it is a pure analysis lemma, we give a proof using probability theory.

Lemma. Let Xn be a sequence of real numbers such that e iu xn converges as n tends to oo for almost all u E R Then X n converges to a finite limit.

We will verify the following Cauchy criterion: Xn converges if for any increasing sequences n k and m k , then lim k-+oo Xnk - X m k = 0. Let U be a random variable which has the uniform distribution on [0, 1 ]. For any real t, by hypothesis a.s. eit Uxn k and e it Ux=k converge to the same limit. Therefore,

Proof.

limoo e it U ( xn k - x=k ) k->

=

so that the characteristic functions converge,

1

a.s.

it (xnk - x= k) U } = lim {e E k ---+ 00

1

,

for all t E R Consequently (xn k - Xm k )U converges to zero in probability, 0 whence limk->oo X n k - Xm k = 0, as claimed.

We will henceforth always assume that we are using the ( unique) dtdlag version of any given Levy process. Levy processes provide us with examples of filtrations that satisfy the "usual hypotheses," as the next theorem shows.

22

I Preliminaries

Let X be a Levy process and let Ot :f1 vN, where (:f1)o t 9u · Note that since the filtration Q is increasing, it suffices to show that 9t nn -> Qt+ -n1. . Thus, we can take countable limits and it follows that if s 1 , . . . , Sn < t, then for ( U 1 , · · · , Un ) =

Theorem 31.

=

=

E{ ei(ur X.l + ··- +un Xsn l l Ot }

=

l

E{ e i(ur Xsl +···+un Xsn ) l Ot+ } + = ei ( ur X s1 +··· u n X sn ) . =

For V1 , . . . , Vn > t and ( u 1 , . . . , Un ), we give the proof for n = 2 for nota­ tional convenience. Therefore let z > v > t, and suppose given U 1 and U z . We have E{ ei(u rXv +u2Xz ) I Ot+ }

lim E{ ei (u rXv +u 2Xz) I Ow } wlt eiu 2Xz i r = lim E{ e u Xv ( uz ) l 9w } z wlt ( fz U z eiu2 Xv r = lim E{ ei u Xv fv ( u z z (U z ) IOw }, wl t =

/ /

· g t hat MU2 usm v

=

• le. Combming eiu2 Xv is a martmga " terms the above becomes fv (u 2 )

and the same martingale argument yields = =

=

+ uz )fz - v (uz ) wl e i (u r+ u 2 )Xt fv - t( U I + U2 )fz - v (U 2 ) limt ei (u r +u )Xw fv - w (U I 2

E { ei(u r Xv +u2X z ) I Ot }.

It follows that E{ ei EuJ X•J I Ot+ } = E{ ei EuJ X• J IOt} for all (s 1 , . . . , sn ) and all (u 1 , . . . , un ), whence E{ ZIOt+ } = E{ ZIOt} for every bounded Z E Vo <s <= :Pj . This implies Ot + = Ot except possibly for events of probabil­ ity zero. However since both a--algebras contain N, we conclude 9t+ = Ot for each t > 0. D The next theorem shows that a Levy process "renews itself" at stopping times.

4 Levy Processes

23

Let X be a Levy process and let T be a stopping time. On the process Y (Yt)o
=

=

E

lA exp {i

L Uj( Xr+tj - Xr+tj-t )} j

=E

= P(A) IT ftj - tj _ ( ui ) j

1

by applying the Optional Sampling Theorem (Theorem 16) n times. Note that this shows the independence of Yt = Xr+t - Xr from :Fr as well as showing that Y has independent and stationary increments and that the distribution

of Y is the same as that of X.

If T is not bounded, we let rn = min(T, n) = T 1\ n. The formula is valid for A n = A n {T < n } when A E :Fr, since then An E :Fr /\n · Taking limits and using the Dominated Convergence Theorem we see that our formula holds for unbounded T as well, for events A = A n {T < oo }, A E :Fr. This gives the result. 0 Since a standard Brownian motion is a Levy process, Theorem 32 gives us a fortiori the strong Markov property for Brownian motion. This allows us to establish a pretty result for Brownian motion, known as the reflection principle. Let B = (Bt)t>O denote a standard Brownian motion, Bo = 0 a.s. , and let St = SUPo< sO be standard Brownian motion (Bo = 0 a.s.) and St = SUPo<s 0, z > 0,

P ( Bt < z - y; St > z) = P (Bt > y + z). Proof. Let T = inf{ t > 0 Bt = z }. Then T is a stopping time by Theorem 4, and P(T < oo) = 1 . We next define a new process X by :

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 B and - 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; Bt < z - y), ( R, X) and (T, B) have the same distribution. However we also

since that R = T identically, whence

have

{ R < t; Xt < z y} = {T < t; Bt > z + y} by the construction of X. Therefore P ( T < t; Bt > z + y) = P (T < t; Bt < z - y). -

The lef t side of ( ) equals *

P(St > z; Bt > 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; Bt < z - y). Combining these yields P(St > z; Bt < z - y ) = P (Bt > z + y), *

0

which is what was to be proved.

We also have a reflection principle for Levy processes. See Exercises 30 and 31. Corollary. Let B = (Bt)t>O be standard Brownian motion St = SUPo <s 0,

(Bo = 0 a.s. ) and

P(St > z ) = 2P(Bt > z ) .

Proof

Take y

=

0 in Theorem 33. Then

P(Bt < z; St

> z ) = P(Bt > z ) .

Adding P(Bt > z ) to both sides and noting that z } yields the result since P(Bt = z ) = 0.

{Bt > z } = { Bt

> z } n { St >

0

4

Levy Processes

25

A Levy process is dtdlag, and hence the only type of discontinuities it can have is jump discontinuities. Letting Xt - = limsit Xs, the left limit at t, we define �Xt = Xt - Xt - ,

the jump at t. If supt I�Xt l < C < oo 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 } < oo for a Poisson process N. Theorem 34. Let X be a Levy process with bounded jumps. Then E{IXt l n } < oo for all n = 1, 2, 3, . . . .

Proof

Let C be a (non-random) bound for the jumps of X. Define the stop­ ping times T1

=

inf{t : IXt l > C}

Since the paths are right continuous, the stopping times (Tn ) n > 1 form a strictly increasing sequence. Moreover I�Xr l < C by hypothesis for any stopping time T. Therefore sup8 I XJ'n I < 2nC by recursion. Theorem 32 im­ plies that Tn - Tn - 1 is independent of :Frn - t and also that the distribution of Tn - Tn- 1 is the same as that of T1 . 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 JR. bounded away from 0 (that is, 0 ¢:. A, where A is the closure of A). For a Levy process X we define the random variables

E A}

Tl

=

inf{t > 0 : �Xt

r;; + 1

=

inf{t > T;i : �Xt E A } .

26

I Preliminaries

Since X has dtdlag paths and 0 ¢:. A, the reader can readily check that {T;t > t} E :Ft+ = :Ft and therefore each T;t is a stopping time. Moreover 0 ¢:. A and cadlag paths further imply Tj > 0 a.s. and that limn _,oo TA = oo a.s. We define 00

n=l

and observe that N A is a counting process without an explosion. It is straight­ forward to check that for 0 < s < t < oo,

NtA - N:- E u{Xu - Xv ; s < v < u < t}, and therefore NtA - Nf is independent of :Fs ; that is, NA has independent increments. Note further that N/ - Nf" is the number of jumps that Zu Xs +u - Xs has in A, 0 < u < t - s. By the stationarity of the distributions of X, we conclude N/ - Nf" has the same distribution as Nl- s· Therefore N A is a counting process with stationary and independent increments. We conclude that NA is a Poisson process. Let v (A) E{Nf} be the parameter of the Poisson process N A ( v ( A ) < oo by the proof of Theorem 34). Theorem 35. The set function A NtA (w) defines a a--finite measure on lR \ {0} for each fixed (t, w) . The set function v (A ) E{Nf} also defines a a--finite measure on lR \ { 0}. Proof The set function A NtA (w) is simply a counting measure: p, (A ) { number of s < t .!::l. Xs (w) E A } . It is then clear that v is also a measure. 0 =

=

f---+

=

f---+

:

Definition. The measure v defined by

E{Nf} E{ L I A (f:::l.Xs) } O<s< l is called the Levy measure of the Levy process X. v ( A)

=

=

We wish to investigate further the role the Levy measure plays in governing the jumps of X. To this end we establish a preliminary result. We let Nt (w, dx) denote the random measure of Theorem 35. Since Nt (w, dx) is a counting measure, the next result is obvious. Theorem 36. Let A be a Borel set of IR, 0 ¢:. A, f Borel and finite on A.

Then

1A f(x)Nt (w, dx)

=

L f( i:::l.X8)1 A (f:::l.X8). O<s< t

Just as we showed that NtA has independent and stationary increments, we have the following consequence.

4 Levy

Corollary. Let A be a Borel set of IR with 0 ¢: A, and let finite on A. Then

i f(x)Nt( · , dx)

is a Levy process.

Processes

f

27

be Borel and

For a given set A (as always, 0 ¢: A), we defined the associated jump process to be Jf = 2:::: �Xsl A ( �X. ).

O< s
By Theorem

36 and its corollary we conclude that Jf

=

i xNt(·, dx).

Hence, Jf is a Levy process itself, it is defined, and Jf < oo a.s. , each t > 0. Theorem 37. Given A, 0 ¢: A, the process Xt - JtA is a Levy process.

Proof

It is clear that we need only check the independence and stationarity of the increments. But

Xt - Jf - (Xs - J: ) = Xt - Xs - 2:::: �Xu l A (�Xu ) s< u
�a = Xt - 2:::: �Xsl{ I�X . I > a } > O<s
for some constant a > 0. The advantage of doing this is that y a then has jumps bounded by a, and hence has finite moments of all orders (Theorem 34). We can choose any a > 0; we arbitrarily choose a = 1. Note that

It _- Xt Jt(-oo,-l]U[l,oo) = Xt - Jr > XNt (·, dx) . lxl l

·v·l

_

The next theorem gives an interpretation of the Levy measure as the expected rate at which the jumps of the Levy process fall in a given set. Theorem 38. Let A and let fl A E L2(dv) .

be Borel with 0 ¢: A. Let v be the Levy measure of X, Then

i

E{ f(x)Nt(·, dx)} = t

i f(x)v(dx)

28

I Preliminaries

and also Proof.

First let f =

Lj aj 1 Aj , a simple function. Then A E{2:>1Nt 1 } = :�:::::>j E{ N/1 } j = t L a1 v ( A1 ) , j

since N/i is a Poisson process with parameter v ( A1 ). The first equality follows easily. For the second equality, let Mf = Nf' -tv(Ai ) · The Mf are LP martingales, all p > 1 , by the proof of Theorem 34. Moreover, E{ Mf } = 0. Suppose A i , AJ are disjoint. We have

E{MfMl } = E {l: )ML+ l - ML ) 2:)M1t+ I - M/t )} k

i

0 = to < t 1 < · · · < tn = t. Using the martingale property E{MjM/ } = E{ L(ML+I - ML )(M/k+I - M/J }.

for any partition we have

k

Using the inequality

lab I < a2 + b2 , we have

2)ML+r - MfJ (ML+r - M/J < i:: ) ML+r - MjJ 2 + l: ) ML+r - M/J 2 . k

k

k

However Lk( Mfk+ r - MfJ 2 < ( N/!' ) 2 + v(Ai )2t2; therefore the sums are dominated by an integrable random variable. Since Mj and M/ have paths of finite variation on [0, t] it is easy to deduce that if we take a sequence ( 7rn ) n > l of partitions where the mesh tends to 0 we have

L (ML+I - Mfk )( ML+ r - M/J = L D.M;D. M{ nl!_.� t tk+ O<s
E{ MfMl} = E{ L D.M; D.M1} = 0; O<s
the expectation above is 0 because Ai and A1 are disjoint, implying that M i and M1 jump at different times. The second equality is now easy to verify for simple functions. For general f, let fn be a sequence of simple functions such that fn l A converges to f in L 2 ( dv ) , and the result follows. 0

4

Levy Processes

Remark. The first statement in Theorem 38 remains true if f 1 A E See Exercise 28. Corollary. Let Then

f : JR.

---*

29

L 1 (dv).

JR. be bounded and vanish in a neighborhood of 0.

100

E{ L f(�Xs)} = t f(x)v (dx). oo O< s
Proof

We need only combine Theorem 38 with Theorem 36.

0

Let A 1 , A2 be two disjoint Borel sets with 0 � A 1 , 0 � A 2 • Then the two processes

Theorem 39.

L �Xs1A1 (�Xs) O<s
J 1 and J2 are Levy processes. To show they are independent, we begin by forming for u , v in JR.,

. J'' ew cut = iuJ,' - 1 E{ e } .e J2 w Dtv - 1. v J'f i E{ e } Then cu and Dv are both martingales, with E{cn = E{Dn = 0. As in the proof of Theorem 38, let ?rn : 0 = to < t 1 < < t n = t be a sequence of partitions of (0, t] with limn mesh(nn ) = 0. Then t

-

·

·

·

E{c� Dn = E{ L:( c�k+l - c�J L(D�t+ t - D�J } k

i

= E{ L( c;;, + , - c�J (D�k+' - n�J}. k

Since cu and Dv have paths of finite variation on compacts, it follows by letting mesh(nn ) tend to 0, that

E { c� Dn = E{ 2:: � c:� D�}. O<s
30

I Preliminaries

which in turn implies, because of the independence and stationarity of the increments that 12 - Ji1 ) +·-+u n ( Jt1n -Jt1n - )) i (v1 ft +··-+vn (Jin -Jin- 1 )) t 2( J + J i(u u 1 1 e t J1 e } E{ = E{ e iu 1Jt\ +O:=j=2 Uj (Jtj -Jt� _1 ) } E{ e ivr Ji1 +i I:j_2 (Jii -Jii-l ) }. Vj

0

This is enough to give independence.

The preceding results combine to yield the following useful theorem, which is one of the fundamental results about Levy processes.

Let X be a Levy process. Then Xt = 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<s l } · Since X has cadlag paths, for each fixed w the function s- r-t X8 (w ) has only finitely many jumps bigger than one on [0, t]. Therefore J has paths of finite variation on compacts. J is also a Levy process by Theorem 36 and its corollary. The process W = X J is also a Levy process (Theorem 37) , and W has jumps bounded by one. We know therefore that, n > 1, E{I Wtl n } exists (Theorem 34), and the stationary increments of W implies E{Wt} = at, for a = E{Wl }. (Recall E{Wo} = 0.) We set yt = Wt - E{Wt } · Then Y has independent increments and mean 0; it is a martingale. Setting Zt = Jt + at completes the proof. 0 While Theorem 40 is the most important result about Levy processes Theorem 40.

=

-

from the standpoint of stochastic integration, the next two theorems provide a better understanding of Levy processes themselves. Theorem 41. Let X be a Levy process with jumps bounded by a. That is, sup8 j � Xs l < a a.s. Let Zt = Xt - E{Xt} · Then Z is a martingale and Zt = Zf + Zf where zc is a martingale with continuous paths, zd is a martingale,

Zf = f x(Nt(-, dx) - tv(dx)), }{lxi < a } and zc and zd are independent Levy processes. Proof. Z has mean zero and independent increments so it is a martingale, as well as a Levy process. For a given set A we define

l xNt (· , dx) - t i xv(dx) = L �Xs 1 A ( �Xs ) - t J xv(dx). A O<s
M/ =

4 Levy Processes 31 For this proof we take a = 1. Let Ak = { k!l < lxl < k }. Then MA • are pairwise independent Levy processes and martingales (Theorem 39). Set Mn = L:::�= l M A • . 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(M[') < Var (Zt) < oo for all n. We deduce that Mr is Cauchy in L2 and hence converges in L2 as n tends to oo to a martingale Zl, 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 M n and Z - M n , for every n. 0

Note that a consequence of the convergence of M[' to Zf in L 2 in the proof of Theorem 41 is that the integral fr- l ,O)u(O,l] x2v(dx) is finite. Note that this improves a bit on the conclusion in Theorem 38. We recall that for a set A, 0 � A, the process Nf = fA Nt(·, dx) is a Poisson process with parameter v(A), and thus Nf - tv(A) is a martingale.

Definition. Let N be a Poisson process with parameter >.. Then Nt - >.t is called a compensated Poisson process. Theorem 4 1 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{ eiu Z� } = e ta 2 u 2 1 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 -

{}{lxl l } }{lxl> l} x(Nt (-, dx) - tv(dx)) + at + L L�.Xs 1 { 1 �X.I > l} = Bt + 1 {lxl < l } O < s
Xt = Bt +

1

where B is a Brownian motion; for any set A, 0 � A, Nf = fA Nt(·, dx) is a Poisson process independent of B; Nf is independent of N[ if A and r are disjoint; Nf has parameter v(A); and v(dx) is a measure on lR \ {0} such that f min ( 1 , x 2 ) v ( dx ) < oo .

Finally we observe that Theorem 42 gives us a formula (known as the Levy-Khintchine formula) for the Fourier transform of a Levy process.

32

I Preliminaries

Theorem 43.

Let X be a Levy process with Levy measure v. Then

where 'lj;(u) = IJ22 u2 - io:u+ }{lr x l > l} (1 - eiux )v(dx) + }{lr xl < l} (1 - eiux + iux )v(dx). o:, the corresponding Levy process is unique in distri­ Moreover given v, 2, IJ bution.

The Levy-Khintchine formula has another, more common expression. ( For a complete discussion of the various formulae of Levy and Khintchine for the case of infinitely divisible distributions, see Feller [72, pages 558-565] .) Indeed, in the n-dimensional case we can express the Levy-Khintchine formulas as follows ( letting ) denote the standard inner product for

]Rn ). (, Theorem 44. Let X be a Levy process in JRn , with X0 0 . Then there exists n JR such a convolution semigroup of probability -measures on that C(X J. l t) t· t The characteristic function of J.lt is e 'I/J (u) , where 'lj; is given nin terms of a v on ]R such that positive definite n n matrix E and a JRn and bounded away from 0; on if A is Borel ((iii) v(A) 1
·

=

=

Levy measure

x

=

centering junction

=

3

'TJ

:

-+

-+

E{O<s< 2:::: f(�Xs)} t 1 = f(x)v(dx). t Second, in the case n 1, one often takes rJ(x) H:"x 2 , possibly at the cost of changing the constant o:. In the next section ( Sect. 6) we discuss semigroups of Markov processes. The semigroup Pt for the Levy process X has the form Pt f(x) J f(x + Y)J.Lt(dy), and one can show innfact that the "infinitesimal generator" of the semigroup Pt is, for f E bC2(lR ), =

=

=

-

oo

=

4

Levy Processes

33

P Af(x) i-tO t f(x)t- f(x) n j = - (V' f(x), a) - "' CTj k fJ2 (x) j,�k= 1 x · Xk + J f(x + y) - f(x) - (V' f(x), ry (y)) v(dy). Note further that if f is constant in a neighborhood of x, then P f f (x (x ) ) t li m i-tO t = j f (x + y) - f (x)v (dy ). = lim

f)

J

f)

We next consider some examples of Levy processes. For the examples we take n = l. has continuous paths, then is identically Example. If a Levy process 0, and must be Brownian motion with drift. That is, = cr + for are the only Levy processes with continuous paths. constants

X

v Xi Bi at, X a, Example. Assume that v is a finite measure. That is, v(lRn ) < and that cr 2 = 0. Then X is a compound Poisson process with jump arrival intensity >. = v(lRn ). That is, let Ni = I::� 1 1{i> T; } be a Poisson process with arrival intensity >. . Let Zi = I:: � 1 Ui l { i>T; } where the sequence (Uj )j>1 are i.i.d., independent of the sequence (Tj )j >l, and .C (U1 ) = � v. Then one easily verifies E{e iuZ, } e - i' 0 and a = at. More precisely, the density for Xi is fi(x) = rl::) x a -1 e - f3 x l{x >O} · In this case cr 2 = 0, and taking a = f3 = 1 gives c/Jt(u) E{e iuZ, } (1 - iu) - i . (See, for example, page 43 of Jacod­ Protter [109] or p. 73 of Bertoin [12].) Then c/Jt (u) = e - i ln ( 1 -iu ) = et.P (u ) . But = 1 , i 'lj;'(u) = i ( 1 - zu. ) = icfJI (u) = iE{e uZ } Jro eiuxe -x dx. Integrating yields = eiux - 1 -x 'lj;( u) }0r x e dx which implies, by the Levy-Khintchine formula, that v( dx) = ! e -x 1{x > O }dx. cr ,

oo,

=

=

=

cpk

E

oo

=

=

=

i

=

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

Sn -- I:�

1

'Yn

xi - f3 n,

'Yn > o ·

X

We say that a random variable has a stable law if it is the limit in distribution as -+ oo 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 o:, 0 < o: < 2, called the index of the stable law, and i X = also constants m 1 > 0, m2 > 0, and 8 such that and

n

E{e u } e'I/J(u )

symmetric n dx) dx ,

One says that a stable distribution is if m1 = m 2 . By analogy we define a symmetric stable process (in the case = 1 ) to be a Levy process with 0 < o: < 2 , where the Levy measure has the form v( = lxld+a) and the constant tJ2 in the Levy-Khintchine formula is 0, which, of course, implies that no Brownian component is present in the process. Note that such stable processes have nice If o: is the index, then the law of is the same as the law of Consult Bertoin [12] for many other properties of stable processes.

(Xt ) t> O

scaling properties. ((3( - � ) Xpt)t> O ·

5 Why the Usual Hypotheses?

We defined the usual hypotheses at the very beginning of this book, and we saw with Levy processes that the natural filtration of a Levy process, once completed, satisfies right continuity, and thus the usual hypotheses hold (see Theorem 31 ) . This phenomenon is true much more generally. Indeed it is true for all "reasonable" strong Markov processes, as well as for all count­ ing processes (cf. , Theorem 25) . In this section we will show that it is true for a very large class of Markov processes. We first begin with some definitions.

a fortiori

X be an JRd-valued process, with X adapted to a filtration IF = (Ft )o < t < oo · X is Markov with respect to IF if Ft and !J{X8; s > t} are conditionally independent, given Xt . Definition 2 . The process X in Definition 1 is Markov with respect to IF if for any random variable Y E btJ{X8 ; s > t} , E{YI Ft } = E{Y I Xt }, a.s., all t > 0. Definition 1 . Let

The following theorem has a straightforward proof.

Definitions and 2 are equivalent. Moreover X is Markov with respect to IF if and only if E{f(Xu )I Ft } = E{f(Xu ) I Xt } a.s. for all u > t > 0 .

Theorem 45.

1

5 Why the Usual Hypotheses?

35

X IF. Markov transition function (Ps ,t )s ,tEIR+

X

Let be a cadlag process and let be Markov with respect to Since X on (!Rd , B) , is Markov, we can associate with it a of Markov where B denotes the Borel sets of JRd . This is a family kernels on (!Rd , B) such that

(s, t, u) with s < t < u. We then have the fundamental rela­ E{f(Xt )I Fs} = Ps,t (Xs, f) a.s. Note that, by using indicator functions such as l A = f, we have P(Xt A I Fs) = Ps ,t (X8, A) a.s.

for each triple tionship

E

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 In this case we have a transition semigroup given by

time

Pt

homogeneous. s t.

for < Definition. Let Co denote continuous functions vanishing at infinity. A Markov process is a Feller process if its transition semigroup has the following two properties. E C0 . (a) For each E Co and each > 0, as 0. (b) For each f E Co and each E JRd , 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. X

(Pt )t>O

f

t PdP f(x) -+ f(x) t -+ x t

Let be a cadlag Feller process(Ffor )O 0 be fixed, and assume f Co. Let Yt = f(Xt )· For 0 < s < t , let E

s t, E{f(X F } ) s I + t E{f(Xt )I Fs+ } E{ f(Xt ) I Xs},

Pt f

Then Y is a martingale for 0 < < and it is cadlag, because is continuous, and X is cadlag. Therefore, Y is a martingale with respect to IF+ too, which means that = Ys, from which we deduce that = 0 and we are done. 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 P�' as a probability measure such that X is Markov with respect to P�' and also .C(Xo) = J.L under P�' , where .C(Xo) denotes the distri­ bution, or "law," of Xo. Let = be the filtration where de­ notes the P�' completion of .Jil, .Jil = o- X : , and fD = o- { Xs : > = Also, let n{IIJJ.I I = }

IF�' (:Ff)o t it is just Theorem 46. Then contains all H of the form H [J� fi (X8 J , for any f E bB. Using the Monotone Class Theorem, 1i ::) bP. By completion, 1i b:F�'. Therefore if h E :Ff+ , we have E{h i :Ft } E{h i :Ff+ } h, P�'­ a.s. s

=

=

=

'H

=

::)

1

=

=

D

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 However it is simply two elementary properties which yield the right continuity of the filtration in a general setting.

Hunt

processes6.

IF,

Definition. Let X be a cadlag Markov process for a filtration which we take to be completed and therefore can assume to be right continuous, in view of Theorem 4 7. 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

T.

Borel.

every stopping time

Suppose now that the semigroup ( Pt ) is That is, if f E B, then Ptf E B as well. Suppose that for every probability measure J.L on (1Rd, B), there exists a probability space ( D, 9, Q) and a cadlag Markov process defined on this space with = J.L . Suppose further that every such cadlag Markov process Y is strong Markov. Then the filtrations = and can be shown to be right continuous in a manner analogous = to the previous proofs.

IF (:Ft )o< t< oo 6

.C(Xo)

See, for example, Sharpe [215, page 219] for

a

(Yt) t> O IF�' (:Ft )o
definition of a Hunt process.

6 Local Martingales

6

37

Local Martingales

Recall that we assume as given a filtered probability space (n, F, lF, P) For a process X and a stopping time T we further recall that X T denotes the

fying the usual hypotheses.

satis­

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 = oo a.s. such that Xt i\ Tn 1 { Tn >O} is a uniformly integrable martingale for each Such a sequence (Tn ) of stopping times is called a fundamental sequence.

n.

n).

Example. Clearly any cadlag martingale is a local martingale (take Tn = Example. We give an example of a local martingale which is not a martingale. Let (Bt)o 0 J!BtJ I < 1/ } . Outside of the ball of radius 1/ centered at the origin the function is harmonic. Again by Ito's formula one can show that this implies u(Bti\Tn ) is a martingale. Since u(Bti\Tn ) is bounded by n it is uniformly integrable. On the other hand, if B0 = x -I 0 then limt--+ oo E{ u(Bt)} = 0 as is easily seen by a calculation, while E{u(B0)} = J j x j - 1 . Since the expectations of a martingale are constant, u(Bt) is not a martingale. (For more on this example, see Sect. 6 of Chap. II, following Corollary 4 of Theorem 27, as well as Exercise 20 in Chap. II.)

u(y)

y

-

=

:

n

u

n

The reason we multiply Xti\Tn by 1 { Tn >O} is to relax the integrability condition on X0 . 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 MT is a uniformly integrable martingale.

Let M, N be local martingales and let S and 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 reduceT M, Tthen S T also reduces M. (d) The processes M , M 1 {T>O} are local martingales. (e) Let X be a cadlag process and Tnlet Tn be a sequence of stopping times increasing to a. s . such that X 1 {Tn >O} is a local martingale for each n. Then X is a local martingale. Proof. (a) follows from the Optional Sampling Theorem (Theorem 1 6 ) and Theorem 13. For (b) , if (S ) > , (T ) >1 are fundamental sequences for M Theorem 48.

V

00

nn 1

nn

38

I Preliminaries

and N, respectively, then Sn 1\ Tn is a fundamental sequence for M + N. For (c) , let Xt = Mt - Mo. Then x svT = X8 + X T - X 8 1\T is a uniformly integrable martingale. But which is in L 1 ' therefore x sv T +Mol {S V T>O} = M S V T I {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 MT I { Tr> O} by the Optional Sampling Theorem (Theorem 16). For (e) , we have x n l { Tn >O} = Mn is a local martingale for each For fixed we know there exists a fundamental sequence un , k increasing to 00 a.s. as k tends to oo. For each choose k = such that P ( Un , k (n ) < Tn 1\ < 2- n . Then limn un , k (n ) = oo a.s., and un , k (n ) 1\ Tn reduces X for each We l take Rm = max(U l , k{ ) 1\ T1 , . . . , um, k (m) 1\ Tm ), and each Rm reduces X by (c) , the Rm are increasing, and lim Rm = oo a.s. Therefore X is a local martingale. D

n.

n,

k(n)

Corollary. Local martingales form a vector space. We will often need to know that a reduced local martingale, and not simply uniformly integrable.

n n)

n.

MT , is in LP

Definition. Let X be a stochastic process. A property 1r is said to hold locally if there exists a sequence of stopping times (Tn ) n -> l increasing to oo a.s. such that X Tn l { Tn >O} has property each > 1. 1f ,

n

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 and Theorems 49 and 50 are consequences of Theorem 48(e) . Theorem 49. X

locally bounded locally square integrable. quare integrable martin­ locally a Let s be a process which is 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. 50.

Let M be adapted, cadlag and let (Tn ) n > l be a sequence of stopping times increasing to oo a.s. If M Tn l{ Tn > O} is a �artingale for each n, then M is a local martingale. Theorem

It is often of interest to determine when a local martingale is actually a martingale. A simple condition involves the maximal function. Recall that Xt = SUPs< t I Xsl and X * = SUPs I Xsl · 5 1.

Let X be a local martingale such thatthen E{Xt } < oo for every tintegrable > 0. Then X is a martingale. If E{X* } < oo, X is a uniformly martingale. Theorem

7 Stieltjes Integration and Change of Variables

39

Proof. Let (Tn ) n > l be a fundamental sequence of stopping times for X. If < t, then E{Xu,rJF. } Xsi\ Tn · The Dominated Convergence Theorem yields E{XtiF.} = X If E{X* } < oo, since each X < X*, it follows that s

=

I tl

• .

(Xt)t >o is uniformly integrable.

D

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 = (At )t>o be a cadlag process. is an increasing pro­ cess if the paths of t At(w) are non-decreasing for almost all is if almost all of the paths of are of called a finite variation process finite variation on each compact interval of JR.+ . Let be an increasing process. Fix an such that At(w) is right continuous and non-decreasing. This function induces a measure on JR.+ . If f is a bounded, Borel function on JR.+ , then J� is well-defined for each > 0. We denote this integral by J; If is bounded and jointly measurable, we can define, w-by-w, the = integral is right continuous in t and jointly = J� measurable. Proceeding analogously for an process (except that the induced measure can have negative measure; that is, it is a we can define a jointly measurable integral

A

A

:

-->

(FV)

A

F8 F(s,w) I(t, w) J.tA (w,ds)

A

t

w

t F(s,w)dA.(w). I A FV

I (t,w) = lo F(s,w)dA.(w) for F bounded and jointly measurable. Let be an FV process. We define

2n I Ait = sup n> l L -

Then

k=l

l A � - At( k -1) I · 2

f---t

J.tA (w, ds) f(s)J.tA (w, ds) f(s)dA.(w).

signed measure ),

t

A

w. A A

2n

IAit < oo a.s., and it is an increasing process.

40

I Preliminaries

Definition. For A an FV process, the total variation process, I A I = (IA i t)t > 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)dA8(w) exists and is finite for all t > 0, a.s. We let t (F A)t(w) = F(s, w)dA8(w)

lo

·

and we write F · A to denote the process F · A = (F · At)t>O · We will also write J; Fs ldAsl for (F · IAI)t. The next result is an absolute continuity result for Stieltjes integrals.

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, oo)) such that 0 < H < 1 and A=H·C or equivalently t At = 1 Hs dC8•

Theorem 52.

Proof

If J.L and

v are two Borel measures on JR.+ with J.L « v, then we can set tt( ( s , t] if v((s, t]) > 0, s,t v(( ' o:(s, t) =

{

otherwise.

0,

Defining h and k by h(t) = liminfsrt o:(s, t), k(t) = lim supt l s 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 = k dv.

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 g = 0, it suffices to define A(rt,w)) (A(t, w) . . . l m m H(t ' w ) = l f '

(C(t, w) - C(rt, w)) such an H is clearly adapted since both A and C are. Corollary. Let A be an FV process. There exists a jointly adapted process H, - 1 < H < 1, such that IAI = H · A and A = H · IAI rT l ,rE
or equivalently and

0

measurable,

7

Stieltjes Integration and Change of Variables

41

We define At = �( I A i t +At) and A; = �( I A i t -At)· Then A + and A­ Proof are both increasing processes, and I A I - A + and I A I - A - are also increasing processes. By Theorem 52 there exist processes H+ and H - such that At = J; H: l dAsl , At = J; H; j dAsl -· It then follows that+ At = At- - A; = J;(H:­ Ht Ht - Ht and suppose H and H are defined as in H;)proof ·ofLetTheorem d Asl I 52. Except for a P-null set, for a given w it is clear that the H · A, we have I Hs(w)l 1 dA8(w) almost all s. Considering t H8dAs = lot H8d(lo8 H l dA 'l ) lo u u t t 1 Hs Hsl dAs l = 1 1 j dAsl = I A i t · =

=

=

This completes the proof.

0

H continuous paths,

When the integrand process has the Stieltjes integral is also known as the Riemann-Stieltjes integral (for fixed 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 [ 95, pages Theorem 53. FV

J; H8dA8

w).

1 316, 317]). an process and let H be a jointly measurable Let A be q H is continuous. Let uence process such that a.s. s be a se (s,w) of finite< Trandom partitions of [0, t] with limn-+oo mesh(7rn ) 0. Then for Tk < Kn

r---+

=

sk

k+ l ,

We next prove a change of variables formula when the FV process is Ito's formula (Theorem 32 of Chap. II) is a generalization of this result. Theorem 54 {Change of Variables). FV

continuous.

Let A be an process with con­ tinuous paths, and let f be such that its derivative f' exists and is continuous. Then (f(At))t>o is an FV process and t f(At) - f(Ao) = lo f' (As)dAs. Proof For fixed w, the function s f' (A8(w)) is continuous on [0, t] and hence bounded. Therefore the integral J; f ' (A8)dA8 exists. Fix t and let be a sequence of partitions of [0, t] with limn-+ oo mesh(7rn ) = 0. Then f(At ) - f(Ao) = t L {f(Atq , ) - f(At k )} k ,tk+! = L f' (Ask )(Atk +! - At k ), k r---+

Kn

E1fn

42

I Preliminaries

by the Mean Value Theorem, for some Sk , tk follows by taking limits and Theorem 53.

< Sk < tk+ l ·

The result now D

Comment. We will see in Chap. II that the sums

L f(At k )(At k + l - At k ) t k E 7rn [O,t] converge in probability to

t

1 J(As- )dAs

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) =

t

10 f' (As_)dAs + O<sl:
The next corollary explains why Theorem 54 is known as a change of variables formula. The proof is an immediate application of Theorem 54. Corollary. Let g be continuous, and let h( t ) = J� g(u)du. Let A be an FV process with continuous paths. Then

h(At) - h(Ao)

=

=

1At g(u)du Aot 1 g(As)dAs ·

We conclude this section with an example. Example. Let N be a Poisson process with parameter >.. Then Mt = Nt >.t, the compensated Poisson process, is a martingale, as well as an FV process. For a bounded (say) , jointly measurable process H, we have

where (Tn ) n > l are the arrival times of the Poisson process N. Now suppose the process H is bounded, adapted, and has continuous sample paths. For 0 < s < t < oo, we then have

8 Naive Stochastic Integration Is Impossible

E{It - IsiFs} = E{ =

t

l HudMu iFs}

limoo E{n--->

""' Ht k (Mt q1 ,t � +l E7rn

43

- Mt k )IFs}

tk k � L E{E {Ht k (Mt k+l - Mt k ) I.Ft k } I Fs} = J� tk, tk+l E7rn - 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 Nalve 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; H8 dA 8 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.

Let X be a Banach space and let Y be a normed linear space. Let {TaJ be a family of bounded linear opemtors from X into Y. If for each x E X the set {Ta x} is bounded, then the set { IITa ll } 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( 7rn ) = 0. We ask the question "What conditions on x are needed so that the sums Theorem 55.

converge to a finite limit as n -+ oo 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.

If the sums Sn of ( ) converge to a limit for every continuous function h then x is of finite variation. Theorem 56.

*

7 The Banach-Steinhaus Theorem is also known as the Principle of Uniform Bound­

edness.

44 Proof.

I Preliminaries

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 h E X, let For each fixed

sign{x(tk + l )

n

Tn (h) = Therefore

h in X such h we have lx(tk + I ) - x(tk )l.

it is simple to construct an - x(tk )}, and llh ll = 1. For such an

II Tn ll

>

I: t k ,t k+l E1fn

I:

t k , t k + l E7rn

that

h(tk )

lx(tk + l ) - x(tk ) l ,

and supn IITn ll > total variation of x. On the other hand for each h E X we have limn-+oo Tn (h) exists and therefore supn II Tn (h)ll < oo. The Banach-Steinhaus Theorem then implies that supn IITn ll < oo, hence the total 0 variation of x is finite. each

n,

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 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 +oo 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� H8dX8 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 [41 J 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 nai"ve stochastic integration is impossible is due to Meyer [1 78] . Exercises for Chapter I

For all of these exercises, we assume as given a filtered probability space (!1, :F, lF, P) satisfying the usual hypotheses. Exercise 1. Let S, T be stopping times, S < T a.s. Show :Fs c Fr . Exercise 2. Give an example where S, T are stopping times, S 1' S is not a stopping time.

<

T, but

-

Exercise 3. Let (Tn ) n > l be a sequence of stopping times. Show that supn Tn , infn Tn , lim sup n-+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 :Fr = nn :Frn . 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{X I :Ft}. -->

(a) Show

Mt E LP, all t > 0.

46

Exercises for Chapter I

(b) Show M is a continuous martingale. Exercise 6. Let N = (Nt)t>o be Poisson with intensity .mteger. Show that E { N - t } 2 (At).X t e-.X t I t A l - ( >.t - )! · _

A. Suppose At is an

1

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 , Bl , Bf) be three dimensional Brownian motion (that is, the Bi are i.i.d. one dimensional Brownian motions, 1 < i < 3) . Let To: = sup t { II Btll < o:}. The time To: is known as a last exit time. Show that if o: > 0, then To: is not a stopping time for the natural filtration of B. Exercise 9. Let (Nf)t>o be a sequence of i.i.d. Poisson processes with pa­ rameter A = 1. Let Mf = ! (Nf - t), and let M = (I:� 1 Mf)t> O · (a) Show that M is well-defined (i.e., that the series converges) in the L2 sense. * (b) Show that for any t > 0, L: s < t LlM8 = oo a.s., where LlM8 = M8 - Ms is the jump of M at time s. Exercise 10. Let (Nf)t>O and (L1)t>obe two independent sequences of i.i.d. Poisson processes of parameters A = 1. Let Mf = � (Nf - LD. (a) Show that M is a martingale and that M changes only by jumps. * (b) Show that L: s < t l llMs l = oo a.s., and t > 0. Exercise 11. Show that a compound Poisson process is a Levy process. Exercise 12. Let Z be a compound Poisson process, with E { I U1 1 } < oo. Show that Zt - E{UI }At 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 a2 = 0 in the Levy­ Khintchine formula) , and a finite Levy measure v. Let A = v(R.). Show that Z is a compensated compound Poisson process with arrival intensity A and i.i.d. jumps with distribution J.L = � v. =

*Exercise 14. Let 0 < T1 < T2 < · · · < Tn < · · · be any increasing sequence of finite--valued stopping times, increasing to oo. Let Nt = I:� 1 1 {t> T; } , and let (Ui)i> 1 be independent, and also the (Ui)i> 1 are independent of the pro­ cess N. Show that if sup i E{IUi l } < oo, all i, and E{Ui } = 0, all i, then Zt = I:� 1 Ui l {t> T;} is a martingale. (Note: We are not assuming that N is a Poisson process here. Furthermore, the random variables (Ui )i> l , 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) = l: akEih (dx), k= 1 where Ef3k (dx) denotes point mass at f3k E lR of size one, with '£"(;' 1 (J�ak < oo. Show that Z is of the form Zt = 2::: f3k (Ntk - ak t) k= 1 00

where (Ntk )t > O is an independent sequence of Poisson processes with param­ eters (ak)k >;· Verify that Z is an L 2 martingale. Exercise 16. Let B = (Bt)o 0.

(a) Show that on [0, t] there are only a finite number of jumps larger than E. That is, if Nt = '£ s < t 1{IA X.I>c: } 1 where b.Xs = Xs - X8_ , then P(Nt < oo) = 1, each t > 0. (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 E > 0. Let .6.Z8 = Zs - Z8_ , and set Jf = L. s < t .6.Zs1{IAZ.I> c:} · Show that Z - J" and J" are independent Levy processes. Exercise 19. Let X be an adapted process with continuous paths. Show that X is locally bounded. *Exercise 20. Give an example of an adapted process X with X0 = 0 that has cadlag paths which are not locally bounded. Exercise 21. (This exercise shows that 2 is the best constant in Doob's quadratic martingale inequality.) Let (0., :F, P) be Lebesgue measure on (0, 1 ) with the Borel sets. Define the filtration

:Ft = {A E :F : (t, 1) c A, or A c (0, t]}, 0 < t < 1 , and :Ft = :F for t > 1 . Intuitively, with this filtration we know all about what is going on during (0, t] , but nothing about what is happening during (t, 1). (a) Show that for Y E L 1 , the martingale Mt = E{Y I :Ft} is given by Y(w), 1 1 Y(s)ds, 1 t t _

1

0 < w < t, t < w < l.

Exercises for Chapter I (b) Let Y (w) = (1 - w) - a for some a, 0 < also if Mt = E{YIFt}, then

48

a < 1/2. Show that Y E £2 and

1 Y(t), t < w < 1. 1 1 (1 - a) (1 - t)"' 1 - a (c ) Show supt Mt = 1 2a Y, w-by-w, and deduce 1 IIYII = 1 M sup MII = I t £2 1 - a II ooll £2· £2 1 - a Note that lima 1 / 2 ( 1 - a) -1 = 2.

Mt (w) --

--+

Exercise 22. (This exercise gives an example of a martingale with each path bounded but which nevertheless is not locally bounded.) Let (n, F, lF, P) be as in Exercise 21. Let Y (w) = .)w, and Mt = E{Y I Ft}. (a) Show that

1 o,t (w) + 2 t (w). Mt ( w) = vW 1( J 1 + Jt 1( , l)

(b) Let T be a stopping time, T not identically 0. Show that T(w) > w on some w-interval (O, c:) with E > 0. (c) Use (b) to show E {Mf.} > J; !ds = oo , and deduce that M cannot be locally an £ 2 martingale. (d) Conclude that M has bounded paths but that Mr 1{ T >O} is unbounded for every stopping time T not identically 0. *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{IMriiFs } < oo, but Ms =!= E{Mr iFs } 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 YT - and Yr -) Let Z be a Levy process, and let Yt = O"{ Z8 ; 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 YT - = O" { A n (t < T) ; A+ E Yt }i at time 0, set Yo - = YO · For a process X, let xr - = Xt1{t< T } Xr- 1 {t>T }, and take Xo- = 0. (a) (b) (c) (d)

Show 9r - = O"{T, zr - } v N. Show YT = O"{T, zr } v N. Show YT = O"{Yr - , Zr } if T is a.s. finite. Conclude that for a bounded time T if Zr = Zr - a.s., then YT

= YT- ·

(Note that if G = Wt)t> O is the completed natural filtration of a Brownian motion, then YT = YT - for all bounded stopping times.)

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 Fr - . Moreover if (Tn ) n> is any increasing sequence of stopping times with limn-+<Xl Tn = T, show that 1

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 = 0. That is, Z and - Z have the same dis­ tribution. Let St = sups
P(St > z ; Zt < z - y) < P(Zt > z + y). Exercise 31. Let Z be a symmetric Levy process with Z0 cise 30. Show that P(St > z) < 2P(Zt > z).

=

0 as in Exer­

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(t k ) = sign (x ( t k+ l) - x(t k )), we need to be able to "see" the trajectory of x on (tk , t k +ll · 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, lF, P) satisfying the

usual hypotheses.

Definition. A process resent ation

H is said to be simple predictable if H has a repn

Ht = Hol{o} (t) + L Hi l(r; ,T;+ 1 J (t) i=l where 0 = T1 < · · · < Tn+ < oo is a finite sequence of stopping times, Hi E Fr; with I Hi I < oo a.s. , 0 < i < n. The collection of simple predictable l

processes is denoted S.

52

II Semimartingales and Stochastic Integrals

Note that we can· take T1 T0 = 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 Su · We also write L0 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 Ix 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 Ix (H). Inspired by the above considerations, for a given process X we define a linear mapping Ix S L 0 by letting :

___,

n Ix (H) = HoXo + 2..: Hi (Xr,+, - XrJ , i=l where H E S has the representation

n Ht = Ho l {o} + L H; l(r, ,T,+tl · 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 cadlag, adapted, and Ix Su L0 is continuous. :

___,

Recall that for a process X and a stopping time T, the notation xr denotes the process (Xt/\T )t> O · Definition. A process X is called a semimartingale if, for each t E [0, oo ) , 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 .

Proof

The set of (total) semimartingales is a vector space.

This is immediate from the definition.

0

2 Stability Properties of Semimartingales

53

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 ?-probability implies convergence Q-probability. 0 Thus the theorem follows from the definition of X. Theorem 3. Let (Pk ) k -> 1 be a sequence of probabilities such that X is a Pk semimartingale for each k. Let R = "£";' >..k Pk , where Ak > 0, each k, and "£": Ak = 1 . Then X is a semimartingale under R as well. Proof Suppose Hn 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 0 Pk. This then implies Ix(Hn ) converges to Ix (H) under R. Theorem 2.

m

1

1

Let X be a semimartingale for the filtmtion lF. Let G be a subfiltmtion of lF, 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 lBI = (1tt)t>o · In this case we have S ( G) is contained in S(JF) . Theorem 4 ( Stricker's Theorem) .

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, lF, P) satisfying the usual hypotheses. Theorem 5 (Jacod's Countable Expansion) . Let A be a collection of events in :F such that if Aa" A,a E A then Aa n A,a = 0, (a =/= f3) . Let 1tt be

the filtmtion genemted by :Ft and A. Then every (lF, P) semimartingale is an (JBI, P) semimartingale also. Proof Let An E A. If P(A n ) = 0, then An and A� are in :Fo by hypothesis. We assume, therefore, that P(A n ) > 0. Note that there can be at most a countable number of An E A such that P(A n ) > 0. If A = Un >1 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 E A. Define a new probability Qn by Qn ( · ) = P( · IAn ) , for An fixed. Then Qn « P, and X is a (lF, Qn ) semimartingale by Theorem 2 . If we enlarge the filtration lF by all the :F measurable events that have Q n ­ probability 0 or 1 , we get a larger filtration Jfn = (:Jt)t>o, and X is a (Jfn , Qn ) semimartingale. Since Qn (A m ) = 0 or 1 for =/= n, we have :Ft C 1tt C :Jt, 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 = L n > 1 P(An ) dQn , m

54

II Semimartingales and Stochastic Integrals

where X is an (IHI, Qn ) semimartingale for each we conclude X is an (IHI, P) semimartingale.

n. Therefore by Theorem 3 0

Corollary. Let A be a finite collection of events in :F, and let IHI = (1tt)t>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 7tt = :Ft 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 X[ - = Xtl{o T } , where Xo - = 0.

Let X be a cadlag, adapted process. Let (Tn ) be a sequence of positive r.v. increasing to oo a.s., and let (Xn ) be a sequence of semimartin­ gales such that for each n, xTn - = (Xn ) Tn - . Then X is a semimartingale. Proof We wish to show x t is a total semimartingale, each t > 0. Define Rn = Tn l{ Tn t}· Then

Theorem 6.

But P(Rn < oo) = P(Tn < t) , and since Tn increases to oo a.s., P(Tn < t) --+ 0 as n --+ oo. Thus if Hk tends to 0 in Su , given E > 0, we choose n so that P(Rn < oo) < c/2, and then choose k so large that P{I Icx n ) < (H k ) l > c} < Ej2. Thus, for k large enough, P{ l lx< (Hk )l > c} < E. o

Corollary. Let X be a process. If there exists a sequence (Tn ) of stopping times increasing to 00 a.s., such that X Tn (or XTn 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

Each adapted process with cadlag paths of finite variation on compacts (of finite total variation} is a semimartingale (a total semimartin­ gale}. Proof. It suffices to observe that llx (H)I < IIHIIu J;' ldXsl, where f000 l dXs l

Theorem 7.

denotes the Lebesgue-Stieltjes total variation and

IIH II u

=

SUP (t , w) IH(t, w) j .

0

Each L2 martingale with cadlag paths is a semimartingale. Proof. Let X be an L2 martingale with X0 = 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 Theorem 8.

n

n

E{(Ix ( H) )2} = E{(I: Hi (Xri+, - XrJ)2} = E{I: Hl(Xr,+, - XrJ2} i=O

<

n

i=O

n

IIHI I�E{'2:: (Xr,+, - XrY } = II H II�E{I:( xf,,+, - xf,, )} i=O

i=O

D

Corollary 1. Each cadlag, locally square integrable local martingale is a semimartingale.

Proof.

Apply Theorem 8 together with the corollary to Theorem 6.

D

Corollary 2. A local martingale with continuous paths is a semimartingale.

Proof.

Apply Corollary 1 together with Theorem

51 in Chap. I.

0

Corollary 3. The Wiener process (that is, Brownian motion) is a semimartin­ gale.

Proof.

The Wiener process Bt is a martingale with continuous paths if B0 is integrable. It is always a continuous local martingale. 0 Definition. We will say an adapted process X with cadlag paths is decom­ posable if it can be decomposed Xt = Xo + Mt + At, where M0 = 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 Xt = 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 0 result. 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 D

able. Theorem 9 then gives the result.

Since Levy processes are prototypic strong Markov processes, one may well wonder if all 1Rn-valued strong Markov processes are semimartingales. Simple examples, such as Xt = BJ I3 , 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 elementary 1 ) . 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) x3), "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 ][]) 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 b!L 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 L0 , the space of finite-valued random variables topologized by convergence in probability. We need to consider a third type of convergence. 1

See Theorem 71 of Chap. IV which proves a similar assertion for Xt 0 < a < 1/2.

4 Stochastic Integrals

57

Definition. A sequence of processes (H n ) n> I converges to a process H uni­ formly on compacts in probability (abbreviated ucp) if, for each t > 0, SUPo<s
Y in ucp We write H! = SUPo< s 0. We write ][])ucp , Lucp , S ucp to denote the respective spaces endowed with the ucp topology. We observe that ][])u cp is a metrizable space; indeed, a compatible metric is given, for X, Y E lDl, by --+

00 1 d(X, Y) = L E{min(l, (X - Y) � ) } . 2n n=l

The above metric space ][])ucp is complete. For a semimartingale X and a process H E S, we have defined the appropriate notion Ix (H) of a stochastic integral. The next result is key to extending this definition. Theorem 10. The space S is dense in L under the ucp topology.

Proof Let Y E L. Let Rn = inf{t : IY't l > n }. Then Rn is a stopping time and yn = yRn l{ 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 = limuu>tt Yu . Then Z E lDl. For E > 0, define TJ = 0 T� +I = inf{t : t > T� and IZt - Zr� l > c: }. Since Z is dtdlitg, the T� are stopping times increasing to oo a.s. as n increases. Let ze = "" 0 n Zr· n l[rn• 'rn·+I ) ' for each E > 0. eThen ze are bounded and converge uniformly to z as E tends to 0. Let u = Yol{o } + � n Zr� l (T�,T�+ l J· Then ue E blL and the preceding implies u e converges uniformly on compacts to Yol {o} + Z_ = Y. -.

Finally, define

n , yn e = Yol{o} + L Zr: l (T[A n ,T[+ 1 " n] i=l

and this can be made arbitrarily close to and n large enough.

Y E bL by taking

c;

small enough 0

We defined a semimartingale X as a process that induced a continuous operator Ix from Su into L 0 . This Ix maps processes into random variables (note that Xri is a random variable) . We next define an operator (the stochas­ tic integral operator), induced by X, that will map processes into processes (note that xri is a process) .

II Semimartingales and Stochastic Integrals

58

Definition. For JX S ][]) by :

H E S and X a cadlag process, define the (linear) mapping

----+

n

Jx (H) = HoXo + 2: Hi(XTi+ l - XT; ) i=l for

H in S with the representation

n

H = Hol {o} + L Hi l (T;,T, + 1 l , i=l Hi E Fr; and 0 = < < · · · < < oo 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.

To T1

Tn+ 1

We use interchangeably three notations for the stochastic integral: ,

Jx(H) =

J Hs dXs = H · X.

Observe that Jx (H) t = Ix t (H). Indeed, Ix plays the role of a definite inte­ gral. For H E S, Ix (H ) = f000 Hs dXs. Theorem 11. Let X be a semimartingale. Then the mapping Jx : Su cp ----+

ID>u cp 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 by Tk = inf{t : I(Hk · X)t l > J}. Then Hk l[o,T k] E S and tends to 0 uniformly as k tends to oo. Thus for every t, P{(Hk · X) ; > J } < P{jHk · Xrk " tl > J } = P{I (Hk l [o , TkJ · X)t l > J } = P {I Ix (Hk l[o,T k AtJ )I > J }

Tk

which tends to 0 by the definition of total semimartingale. We have just shown that Jx : S u ][])u cp is continuous. We now use this to show Jx : Su cp ][])u cp is continuous. Suppose Hk goes to 0 ucp. Let J > 0, > 0, t > 0. We have seen that there exists 'I} such that I IH II u < 'I} implies P(Jx (H); > J ) < � - Let Rk = inf { s : IH:I > 1J}, and set Hk = Hk l [o, Rk ]l{ Rk >O} · Then Hk E S and IIHk ll u < 'I} by left continuity. Since Rk > t implies (Hk · X); = (Hk · X);, we have ----+

c

----+

4 Stochastic Integrals

P((H k · X ); > J ) < P((Hk · X); > J) + P(Rk < < + P((Hk ); > rJ)

�

t)

< c:

if k is large enough, since lim k -H'o P((Hk ); > ry )

=

59

D

0.

We have seen that when X is a semimartingale, the integration operator Jx is continuous on Su cp, and also that Su cp is dense in Lu cp · Hence we are able to extend the linear integration operator Jx from S to L by continuity, since ID>u cp is a complete metric space.

Definition. Let X be a semimartingale. The continuous linear mapping Jx : Lu cp ---> ID>u cp obtained as the extension of Jx S ---> ID> 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 (At ) t >o has continuous paths of finite variation with A0 = 0, then the Riemann-Stieltjes integral of J; A8dA8 yields the formula (see Theorem 54 of Chap. I) :

1t A8dA8 0

=

1

- A2t . 2

Let us now consider a standard Brownian motion B = (Bt)t>o with B0 = 0. The process B does not have paths of finite variation on compacts, but it is a semimartingale. Let (1r ) be a refining sequence of partitions of [0, oo) with limn -+oo mesh(7rn ) = 0. Let Bf = L t .E 11"n Bt . l (t . , t • +tl · Then B E L for each n. Moreover, B n converges to B m ucp. Fix t > 0 and assume that t is a partition point of each 7rn · Then n

JB ( B

n

n

) t = L Bt . (Bt k+ t - B t • ) ttkEk <11"tn

and JB ( B ) t

=

limoo JB (Bn )t = lim n� t k E7rn , t k < t n

= lim n

L { (Bt k+ l + Bt . )(Bt k + l - Bt . ) t kE 11"n , t k
�

where we have used telescoping sums. The last term on the right, however, converges a.s. to t, by Theorem 28 of Chap. I. We therefore conclude that

60

II Semimartingales and Stochastic Integrals

1t B8dBs = -B 1 t 0

2

2

-

1 -t, 2

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 lL. Recall that the stochastic integral defined in Sect. 4 will be denoted by the three notations Jx (H) = H · X = J H8dX8 • Evaluating these processes at t, we have

H · Xt = lr H8dXs = l{ HsdXs · o ro , tJ To exclude 0 in the integral we write

t { HsdXs = { H8dX8 • lco , tJ lo + The integral J000 H8dX8 is defined to be limt --+oo J� H8dX8 when the limit exists. Note that J� H8dX8 = HoXo + J�+ H8dX8• For a process Y E lDl , we recall that �yt = yt - yt _ , the jump at t. Also since Y0_ = 0 we have �Yo = Y0 . Recall further that for a process Z and stopping time T, we let zr denote the stopped process (Zr) = Zt11r. 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 Xt (w ) and t yt (w) are the same functions} = 1. f---'

Theorem

H · (XT ).

12.

f---'

Let T be a stopping time. Then (H · X) T = H1[o,T] X = ·

The jump process �(H·X)s is indistinguishable from H8(�X8). Proofs. Both properties are clear when H E S, and they follow when H E JL by passing to the limit. Indeed, we take convergence in ucp for each t, then Theorem 13.

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 HQ · X is Q indistinguishable from Hp · 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 ?-probability implies convergence in Q-probability. 15.

D

Let Pk be a sequence of probabilities such that X is a Pk semimartingale for each k. Let R = 'L,"; >.. k Pk where >.. k > 0, each k, and 'L,"; >.. k = 1 . Then HR · X = Hpk · X, Pk -a.s., for all k such that >.. k > 0. Proof. If >.. k > 0 then Pk � R, and the result follows by Theorem 1 4. Note D that by Theorem 3 we know that X is an R semimartingale. 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 = (Qt)t>o be another filtration such that H is in both JL(G) and L(F), 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 Theorem 1

1

in the proof of Theorem 10, we can construct a sequence of processes H n converging to H where the construction of the Hn depends only on H. Thus H n E S (G) n S(.IF') and converges to H in ucp. Since the result is clear for S , D the full result follows by passing to the limit.

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 JL-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 pacts, then H · X is indistinguishable from the Lebesgue-Stieltjes integral,

puted path-by-path. Proof The result is evident

com­ com­

for H E S. Let Hn E S converge to H in ucp. Then there exists a subsequence n k such that limnk _.00 (Hn k - H); = 0 a.s., and the result follows by interchanging limits, justified by the uniform a.s. D convergence.

Theorem 18. Let X, X be two semimartingales, and let H, H E lL. Let A = {w : H. (w) = H. (w) and X. (w) = X . (w )}, and let B = {w : t Xt (w ) is of finite variation on compacts}. Then H · X = H · X on A, and H · X is f---'

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) > 0. 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 HQ · X = HQ · 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) > 0. Then R � P and B = n, R-a.s. Hence HR · X equals the Lebesgue-Stieljes integral 0 R-a.s. by Theorem 17, and the result follows by Theorem 14. 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 < T(w)} D = {w : t Xt (w) is of finite variation on S(w) < t < T(w) }. f---'

S XT - X on C and H · XT - H · X S equals Then H · XT - H . X s = H ·H ·a path-by-path Lebesgue-Stieltjes integral on D. Proof Let yt = Xt - Xt11 s . Then H · Y = H · X - H · X 8 , 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 0 to obtain the result.

The stochastic integral process Y = H · X is itself a semimartingale, and for G E L we have

Theorem 19 (Associativity) .

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 (n k ) such that Hn k · X converges a.s. to H · X. Let G E S and let y n k = Hn k ·X, Y = H · X. The y nk 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 = limoo Q . yn k = lim G · (Hnk · X) nk �OO nk � limoo (GHn k ) . X nk � =

which equals limn k -+oo Jx (GHn k ) = Jx (GH), since X is a semimartingale. Therefore Jy (G) = Jx (GH) for G E S.

5 Properties of Stoch tic Integrals as

63

Let en converge to G in Su . Then en H converges to GH in Lu cp, and since X is a semimartingale, t lim n-+oo Jy (en ) = limn -+oo Jx ( Qn H) = Jx ( GH) = Jy (G) . This implies y is a total semimartingale, and so Y = H · X is a D semimartingale.

Theorem 1 9 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 X0 = 0. Also, if T k increases to oo 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 £. Let Hn_E S be such that Hn converges to H in ucp. We can_then modify Hn , call it Hn , such that H n is bounded by f!..L Hn E S, and Hn converges ':_niformly to H in probability on [0, t]. Since Hn E bS, one can check that Hn · X is a martingale. Moreover

kn E{(H n · X) ;} = E('j� (H[' (X[' +1 - xr )) 2 } i =l kn

< £2 E{ �)x,f, + l i= l

- x,f, )}

and hence (H n · 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 JR) 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 : T[/ < T{' < · · < Tkn ·

is said to tend to the identity if (i) limn supk Tf: = oo a.s.; and (ii) Jl un ll sup k JTf:+l - Tf:l converges to 0 a.s. Let Y be a process and let be a random partition. We define the process Y sampled at u to be =

u

Y"" - Y 1 { } + 2::= Yrk 1 (rk ,Tk +l ] · o

o

k

It is easy to check that

f �"" dXs

=

Yo Xo + 2.:: Yr, ( X T'+ 1 - XT ) '

,

•

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 (u n ) be a sequence of random partitions tending to the identity. Then the processes f�+ �""ndXs 2:i Yr;: (X{+ 1 - x'{t ) tend to the stochastic integral (y_) · X in ucp. Proof (The notation Y_ means the process whose value at s is given by (Y_ )s limu-. s, u<s Yu; also, (Y_ ) 0, by convention.) We prove the the­ orem for the case where Y is dtdHtg, the other case being analogous. Also without loss of generality we can take X0 0. Y cadlag implies y_ E JL. Let y k E S such that y k converges to Y_ (ucp). We have =

=

o =

=

5 Properties of Stochastic Integrals

J (Y_ - Y"n )8dXs J (Y_ - yk) sdXs + J(Yk - (Y-t)"n )sdXs + J ( (Y-t)"n - Y"n )s dXs

65

=

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 Lu cp and since Y yk --+ 0, we have J(y_ - Yk)8dX8 --+ 0 (ucp). The same reasoning applies to the third term, for fixed n , as k --+ oo. Indeed, the convergence to 0 of ((Y�)"n - Y"n ) as k oo 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)"n )8dX8 tend to 0 as n --+ oo. 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 0 small for large enough n . _ _

--+

-

We consider here another example. We have already seen at the end of Sect. 4 that if B is a standard Wiener process, then

1t B8dBs 0

=

1 1 -B'f -t, -2 2

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 innocent as it may seem if we want to have both of these properties. Example. Let Mt = Nt - >.t, a compensated Poisson process (and hence a martingale with Mt E LP for all t > 0 and all p > 1). Let (Ti)i> l be the jump times of M. Let Ht = 1[o,T, ) (t) . Then H E ID>. The Lebesgue-Stieltjes integral as

lS

II Semimartingales and Stochastic Integrals

66

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)t>o, is defined by

J

[X, X] = X2 - 2 X_dX

(recall that Xo - = 0) . The quadratic covariation of X, bracket process of X, Y, is defined by

[X, Y] = XY -

Y, also called the

J X_dY - J y_ dX.

It is clear that the operation (X, Y) --+ [X, Y] is bilinear and symmetric. We therefore have a polarization identity

[X, Y] = 21 ([X + Y, X + Y] - [X, X] - [Y, Y] ). 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 cadlag, increasing, adapted process. Moreover it satisfies the following. (i) [X, X]0 = XJ and �[X, X] = (�X) 2 . (ii) If CJn is a sequence of random partitions tending to the identity, then xJ + L (xrt·p - xrt) 2 --+ [X, x] i

with convergence in ucp, where CJn is the sequence 0 = Tf: < Tf < · · · < Tt < · · · < Tf:n and where Tt are stopping times. (iii) If T is any stopping time, then [X T , X] = [X, XT ] = [XT , XT ] = [X, X] T .

3 Ruth Williams has commented to us that this example would be more convincing if were itself a semimartingale, H bounded and in Jl)), and J; H d were not

M

a semimartingale. Such a construction is carried out in [1].

s

M

s

6

The Quadratic Variation of a Semimartingale

67

Proof X is cadlag, adapted, and so also is f X_dX by its definition; thus [X, X] is cadlag, adapted as well. Recall the property of the stochastic integral: �(X_ · X) = X_ �X. Then (�X) ; = (Xs - Xs- ) 2 = x; - 2Xs Xs - + x;_ = x; - x;_ + 2Xs_ (Xs- - Xs) = �(X 2 )8 - 2Xs- (�Xs), from which part (i) follows. For part (ii), by replacing X with X = X - X0 , we may assume X0 = 0. Let Rn = supi Tt . Then Rn < oo a.s. and limn Rn = oo a.s., and thus by telescoping series

(X 2 ) Rn = L {(X2 f:+1 (X 2 )Tt } _

i

converges ucp to X 2 . Moreover, the series L i Xrr (xr::.- 1 -xrt ) converges in ucp to J X_dX by Theorem 21, since X is cadlag. Since b2 - a2 - 2a(b - a) = (b - a) 2 , and since Xr,n (xr::.- 1 - xrt ) = xrt (xrt+1 - xrt ) , 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]8 < [X, X]t a.s., with the null set depending on s and t; it is the property that [X, X] has cadlag paths that allows one to eliminate the dependence of the null set on s and t.) Part (iii) is a simple consequence of 0 part (ii) . An immediate consequence of Theorem 22 is the observation that if B is a Brownian motion, then [B, B]t = 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 < sup I XTi+ 1 i

- x rt I V,

where V is the total variation. Therefore the sums tend to 0 as llcrn ll ----> 0. Theorem 22 has several more consequences which we state as corollaries. Corollary 1. The bracket process [X, Y] 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, Y] 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 Then X Y is a semimartingale and

J

X, Y be two semimartingales.

J

XY = X_dY + y_dX + [X, Y].

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 above. 0 In the integration by parts formula above, we have Hence evaluating at 0 yields

(X_ )0 = (Y_ )0 = 0.

XoYo = (X-)oYo + (y_)oXo + [X, Y]o. Since [X, Y]o = �Xo�Yo = X0Yo, the formula is valid. Without the conven­ tion that (X_ )0 = 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.

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 ern is a sequence of random partitions tending to the identity, then [X, Y] = X0Y0 + lim "' (X TZ+ l - x rt )(Y TZ+ l - yTt ),

Theorem 23.

n--+oo � i

where convergence is in ucp, and where ern is the sequence 0 = T0 < T[' < · · · < �n < · · · < TJ:n , with �n stopping times. (iii) IfT is any stopping time, then [X T , Y] = [X, yT] = [XT , YT ] = [X, Yj T .

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 o:, (3, "'( be functions mapping [0, oo) to lR with o: (O ) = 0 . Suppose o:, (3, 'Y are all right continuous, o: is of finite (3(0) = 'Y ( O) variation, and (3 and 'Y are each increasing. Suppose further that for all s, t =

with 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 and d"f. Note that ldo: l denotes the total variation measure corresponding to the measure do:, the Lebesgue-Stieltjes signed measure induced by o:. 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 H and K be two measurable processes. Then one has a.s.

By Theorem 24 we only need to show that there exists a null set such that for w rJ. N, and (s, t) with s < t, we have

Proof

N,

Let N be the null set such that if w rJ. N, then 0 < J: d[X + rY, X + rY] , for every r, s, t; s < t, with r, s, t all rational numbers. Then u

0<

[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 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 r,

*

.

70

II Semimartingales and Stochastic Integrals

Corollary. Let X and Y be two semimartingales, and let measurable processes. Then

if .! + p

Proof

.! q

=

H and K be two

1.

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 H and K be adapted. Since the process [X, X] is non-decreasing with right continuous paths, and since �[X, X]t (� Xt) 2 for all t > 0 (with the convention that X0_ 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 by-path continuous part of [X, X].

[X, X] c denotes the path­

We can then write

[X, X]t

=

=

[X, X] f + xJ + L (�Xs) 2 O<s
0. Observe that [X, X]0 the path-by-path continuous part of [X, Y]. Comment. In Chap. IV, Sect. 7, we briefly discuss the continuous local mar­ tingale part of a semimartingale X satisfying an auxiliary hypothesis known as Hypothesis A. This (unique) continuous local martingale part of X is de­ noted x c . It can be shown that a unique continuous local martingale part, x c , exists for every semimartingale X. 4 (If X is an FV process, then x c 0, as we shall see in Chap. III.) It is always true that [X c ,x c ] [X, X] c . Although we will have no need of this result in this book, this notation is often used in the literature. We remark also that the conditional quadratic variation of a semimartin­ gale X, denoted (X, X), is defined in Chap. III, Sect. 4. It is also true that (X c , x c ) [Xc, xc] [X, X] c , and these notations are used inter­ changeably in the literature. If X is already continuous and X0 0, then =

=

=

=

=

(X, X) 4

=

[X, X]

=

[X, X] c .

=

See, for example, Dellacherie-Meyer [46, page 355] . See also in this regard Exer­ cise 6 of Chap . IV which proves the result for L2 martingales.

6 The Quadratic Variation of a Semimartingale Definition. A semimartingale

X

will be called

qu

71

adrat ic pure jump if

[X, X]c = 0. If X is quadratic pure jump, then [X, X]t = XJ + l: o<s< t (.6.Xs ) 2 . The Poisson process N is an obvious example of a quadratic pure jump semi­ martingale. From the definition or immediately from Theorem 22, we see that [N, N] t = Nt. More generally it can be shown that if X is a Levy process with a Levy decomposition Xt = Bt + yt as in Theorem 42 of Chap. I, where B is a Brownian motion and

{{jxjl }

yt = }

(using the notation of Theorem 42 of Chap. I), then Y is a quadratic pure jump semimartingale. Last note that the trivial continuous process Xt = t is quadratic pure jump since [X, X]f = [X, X]t 0. The next theorem gives a simple criterion for a process X to be a quadratic pure jump semimartingale. -

If X is adapted, cadlag, with paths of finite variation on com­ pacts, then X is a quadratic pure jump semimartingale. Proof Without loss of generality we assume X0 = 0. We have already seen that such an X is a semimartingale (Theorem 7), and that the stochastic inte­ gral with respect to X is nothing more than a path-by-path Lebesgue-Stieltjes Theorem 26.

integral (Theorem 17) . The integration by parts formula for Lebesgue-Stieltjes differentials applied to X times itself yields X 2 = I X_dX + I XdX, com­ puted path-by-path. The semimartingale integration by parts formula (Corol­ lary 2 of Theorem 22 ), on the other hand, yields X 2 = 2 I X_dX + [X, X]. Moreover

J XdX J (x_ + .6.X)dX J X_dX + J .6.XdX, =

and

t

=

t

t

(.6.Xs) 2 • 10 X_dXs + 10 .6.XsdXs 10 Xs_dXs + l: s
Equating the two formulas we deduce theorem.

[X, X]t

=

Note in particular that if X is adapted with variation, then [X, X]t = XJ, all t > 0.

l: s
continuous

0

paths of finite

Let X be a local martingale with continuous paths that are not everywhere constant. Then [X, X] is not the constant process XJ, and X 2 [X, X] is a continuous local martingale. Moreover if [X, X]t = 0 for all t then Xt = 0 for all t. Theorem 27. -

72

II S emi ma tingales and Stochastic Integrals r

Proof. Note that a continuous local martingale is a semimartingale (Corol­ lary 2 of Theorem 8). We have X 2 - [X, X] = 2 J X_dX, and by the mar­ tingale preservation property (Theorem 20) we have that 2 J X_dX is a local martingale. Moreover �2 J X_dX = 2(X_ )(�X), and since X is continu­ ous, �X = 0, and thus 2 J X_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 Xo = 0. Next assume that [X, X] actually were constant. Then [X, X]t = [X, X]o = X5 = 0, for all t. Since X 2 - [X, X] is a local mar­ tingale, we conclude X 2 is a non-negative local martingale, with xJ = 0. Thus Xf = 0, all t. This is a contradiction. If X0 is not identically 0, we set 0 Xt = Xt - Xo and the result follows. A

The following corollary is of fundamental importance in the theory of martingales. Corollary 1 . Let X be a continuous local martingale, and S < T < oo be stopping times. If X has paths of finite variation on the stochastic interval (S, T), then X is constant on [S, T] . Moreover if [X, X ] is constant on [S, T] n [0, oo) , then X is constant there too.

Proof M = xr - X 8 is also a continuous local martingale, and M has finite variation on compacts. Moreover [M, M] = [X T - X 8 , XT - X s ] = [X, X] T - [X, X] 8 , and therefore [M, M] is constant everywhere, hence by Theorem 27, M must be constant everywhere; thus X is constant on [S, T] . 0 Observe that if t > 0 is arbitrary and we take S = 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 v 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

J

J

XY = X_dY + y_dx + [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 = 0. Thus A - B is a continuous local martingale, A0 - Bo = 0, and it has paths of finite variation on compacts. Corollary 1 yields At - Bt - Ao + Bo = 0 and 0 we have uniqueness. Corollary 2 can be useful in determining the process [X, Y] . For example if X and Y are locally square integrable martingales without common jumps such that XY is a martingale, then [X, Y] = X0Y0 . One can also easily verify (as a consequence of Theorem 2 3 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] = X0Y0 , 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, M]t always exists and is finite a.s. for every t > 0 . We use this fact in the next corollary. Corollary 3. Let M be a local martingale. Then M is a martingale with E{Mn < oo, all t > 0, if and only if E{[M , M]t} < oo, all t > 0. If E{ [M, M]t} < oo, then E{Mn = E{[M, M]t}.

First assume that M is a martingale with E{Mn < oo for all t > 0. Then M is clearly a locally square integrable martingale. Let Nt = Ml [M, M]t = � J: M8_dM8, which is a locally square integrable local martingale by Theorem 20. Let (Tn ) n >l be stopping times increasing to oo a.s. such that Nt n is a square integrable martingale. Then E{Nr } = E{No} = 0, all t > 0. Therefore

Proof

E{Ml,\Tn } = E{[M, M]tA Tn }. quadratic inequality gives E{(Mt ) 2 } < 4E{Mn <

Doob's maximal Therefore by the Dominated Convergence Theorem

E{Ml!\ rn } E{Mt2 } = nlim -->oo = lim E{[M, M]t ATn } n -->oo = E{[M, M]t},

oo.

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} < oo. For the converse, we now assume E{[M, M]t} < oo, all t > 0. Define stopping times by yn = inf {t > 0 : IMtl > n} 1\ n. Then yn increase to oo a.s. Furthermore (MT n )* < n + I�Mrn l < n + [M, M];/ 2 , which is in £ 2 . By Theorem 51 of Chap. I, MTn is a uniformly integrable martingale for each n. Also we have that E{(Mr) 2 } < E{((MTn ) * ) 2 } < oo, for all t > 0. Therefore MTn satisfies the hypotheses of the first half of this n theorem, and E{(Mr) 2 } = E{[MT , Mr n]t } · Using Doob's inequality we have E{(Mt*A Tn ) 2 } < 4E{(MtTn ) 2 } = 4E{[MTn , MTn ]t}

4E{[M, M] rnA t} < 4E{[M, M]t}.

=

The Monotone Convergence Theorem next gives < 4 E{[M, M]t} <

oo.

Therefore, again by Theorem 51 of Chap. I, we conclude that M is a martin­ gale. The preceding gives E{ Ml} < oo. 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] 00} < oo, then M is a square integrable martingale (that is supt E { M?} = E { M!} < oo) . Moreover E{Ml} = E{ [M, M]t} for all t, 0 < t < oo. 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 JR3 with Bo = (1, 1, 1 ) . Let Mt = IIBt ll - 1 , where llxll is standard Euclidean norm in JR3 . (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} < oo for all t. However instead of t r---t E{Ml} being an increasing function as it would if M were a martingale, limt ___, 00 E{Ml} = 0. Moreover E{[M, M]t} > 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} = oo for all t > 0. In conclusion, M = IIB II - 1 is a continuous local martingale with E{Ml} < oo for all t which is both not a true martingale and for which E{Ml} < oo while E{[M, M]t} = oo for all t > 0. (Also refer to Exercise 20 at the end of this chapter.)

6 The Quadratic Variation of a Semimartingale Corollary 5. Let X be a continuous local martingale. Then have the same intervals of constancy a.s.

Proof

X

and

75

[X, X]

Let r be a positive rational, and define

Then M = XTr - x r 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

Br = inf{ t > r [X, X]t > [X, X] r }. :

Then

E{(Xsr - Xr) 2 } = E{X�r } - 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 - Xr) 2 } = 0, and Xsr = Xr a.s. Moreover this implies Xq = Xs a.s. on {Sq = Br} for each pair of rationals (r, q), and therefore we deduce that any interval of constancy of [X, X] is also one of X. o Note that the continuity of the local martingale X is essential in Corol­ lary 5. Indeed, let Nt be a Poisson process, and let Mt = Nt t. Then M is a martingale and [M, M]t = 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<s
q

-

lo

and, in particular,

76

II Semimartingales and Stochastic Integrals

[H · X, H · X]t =

lo H'fd[X, X]8• t

Proof. First assume (without loss of generality) that X0 to establish the following result

[H · X, Y]t =

=

Yo = 0. It suffices

t

1 H8d[X, Y]8,

and then apply it again, by the symmetry of the form [·, ·], 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 [X T , 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, T]l where S, T are stopping times, S < T a.s., and U E :Fs . Then f H8dX8 = U(X T - X 8 ), and by Theorem 23

[H · X, Y] = U{[Xr , Y] - [X 8 , Y] } H8d[X, Y]s · = U{[X, Y f - [X, Y] 8 } =

J

The result now follows for H E S by linearity. Finally, suppose H E IL and let Hn be a sequence in S converging in ucp to H. Let zn = Hn ·X, Z = H ·X. We know zn , Z are all semimartingales. We have f H';d[X, Y] s = [zn , YJ , since Hn E S, and using integration by parts

J J

J

[zn , Y] = Y zn - Y_dzn - zn dY = yz n y_Hn dX - zn dY.

J

By the definition of the stochastic integral, we know since Hn ----> H ( ucp), letting n ----> oo we have

zn ----> Z

in

ucp,

and

J j Y Z - J Y_dZ - J Z_dY

n , Y] = YZ - Y- HdX - z_dY [z n---> oo lim

= =

[Z, Y],

again by integration by parts. Since limn __, 00 J H�d[X, Y]s = J H8d[X, Y]8, we have [Z, Y] = [H · X, Y] = J H8d[X, Y]s, and the proof is complete. 0

Example. Let Bt be a standard Wiener process with Eo = 0, (i.e., Brownian motion). Bl - t is a continuous martingale by Theorem 27 of Chap. I. Let H E IL be such that E{f H'fds } < oo, each t > 0. By Theorem 28 of Chap. I

�

6 The Quadratic Variation of a Semimartingale

[B , B ] t

77

[H · B , H B] t I� I H8dB8 t E{(l H8dB8) 2 } E{[H B , H · B] t } E{ fot H?ds}.

we have t, hence H?ds. By the martingale · preservation property, is also a continuous local martingale, with 0. By Corollary 3 to Theorem 27 ·

(H B )0

=

=

=

=

·

=

It was this last equality,

that was crucial in K. Ito's original treatment of a stochastic integral. Theorem 30. Let be a cadlag, adapted process, and let be

H semimartingales. Let

X, Y two be a sequence of random partitions tending to the identity. Then n t , r yr 1 TI+ l ) x )(Y H (X Tt+ ,n : :: l: r (0 < T0 < T]' < converges in ucp to I H8_d [X, Y] s (Ho - 0) . Here · · · < Tt < · · · < Tf:J . Proof. By the definition of quadratic variation, [X, Y] XY -X_· Y-Y_ ·X, where X_ · Y denotes the process (J� X8_dYs ) t >o· By the associativity of the stochastic integral ( Theorem 19) H_ . [X, YJ H_ . (XY) - H_ . (X_ . Y) - H_ . (Y_ . X) H_ . (XY) - (H_X_) . Y - (H_Y_) . X H_ . (XY) - (HX)_ . Y - (HY)_ . X. an

_

_

=

an

=

=

=

=

=

By Theorem 2 1 the above is the limit of

i

=

- yTt (XTI+ l - XT,n )} 2::::: HTt ( xT,''p - xr,n ) (YTDt-1 - yTt ) i

0

0

78

II Semimartingales and Stochastic Integrals

7 Ito's Formula ( Change of Variables)

A H

Let be a process with continuous paths of finite variation on compacts. If E lL we know by Theorem 17 that the stochastic integral agrees a.s. with the path-by-path Lebesgue-Stieltjes integral J In Sect. 7 of Chap. I (Theorem 54) we proved the change of variables formula for f E C 1 , namely

H8dA8•

We also saw at the end of Sect. process with 0,

B0 =

B

Taking f(x)

HA ·

4 of this chapter that for a standard Wiener

1B 1 . t 8 BsdB = 2 'f - 2 t Jo

x2 /2 , the above formula is equivalent to t t 1 f ( Bt ) - f ( Bo ) = f ( Bs )dBs + -2 f (Bs)ds ,

=

1 0

1 0

I

II

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 thanks to Theorem case of Theorem

32,

26.

Let V be an F V process with right continuous paths, and let f be such that f1 exists and is continuous. Then ( f( Vt ) ) t>o is an FV process and

Theorem 31 (Change of Variables) .

f( Vt ) - f(Vo) =

t !'Ws- )dVs +

Jo

+

J�+

2:.::

( { ) ) ) } s Vs s Vs 'W fW � f · ! st

O< <

= fc o t denotes the integral over the half Recall that the notation ,] open interval (0, t]. We wish to establish a formula analogous to the above, but for the stochastic integral; that is, when the process is a semimartingale. The formula is different in this case, as we can see by comparing equation ( *) with equation ( **); we must add an extra term!

Let X be a semimartingale and let f be a C2 real function. Then f(X) is again a semimartingale, and the following formula holds: Theorem 32 (Ito's Formula) .

7 Ito's Formula ( Change of Variables)

f( Xt ) - f(Xo ) =

79

t f'(Xs- )dXs + � t j"(Xs_ ) d[X, X]� lo + lo + +

I: {f(Xs) - f(Xs- ) - f'(Xs-)�Xs} ·

O <s< t

Note that the jump part of the stochastic integral J f" (Xs-) d[X , X] s is given by Ls < t J"(Xs- )(�X8 )2 , and this is a convergent series. By adding and subtracting 1/2 of this series, we can rewrite Ito's formula in the equivalent form

Proof.

f(Xt)-f(Xo) = +

t d + � t f"(Xs- ) d[X , X] s Xs j'(Xs_) 2 JO+ lo +

I: {J(Xs) - f(Xs-) - J'(Xs-)�Xs - �j" (X8 _)(�X8 ) 2 }

O <s
which is perhaps less obviously a generalization of the "classical" case, but notationally simpler to prove. The proof rests, of course, on Taylor ' s Theorem which says

1 f(y) - f(x) = f (x)(y - x) + 2 f (x)( y - x)2 + R(x, y) where JR(x, y)J < r(Jy - xi)(y - x) 2 , such that r : R. + --> R. + is an increasing function with limulO r(u) = 0, which is valid for f E C 2 defined on a compact I

II

set. We now prove separately the continuous case and the general case. Proof for the continuous case. We first restrict our attention to a continuous semimartingale X , since the proof is less complicated but nevertheless gives the basic idea. Without loss of generality we can take = 0. Define stopping times Rm = inf{t : J Xtl > m } . Then the stopped process X R = is bounded by m , and if Ito's formula is valid for X R= for each m , it is valid for X as well. Therefore we assume that X takes its values in a compact set. We fix a t > 0, and let an be a refining sequence of random partitions5 of [0, t] tending to the identity [an = (0 = Tf) < Tf < . . . < Tf:n = t)]. Then

Xo

f(Xt) - f(Xo ) = L { f (Xr,+ , ) - f (Xr,n ) } kn

i=O

(Xr;:.-1 - Xr,n) + � l:: f" (Xr,n ) (xr,+ 1 - Xr,n f + l:: R (Xr,n Xr,+ 1 )

= I: !' ( Xr,n ) t

i

5

i

,

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 I� f' (X8 _ ) dX8 by Theorem 2 1 ; the second sum converges to � I� f"(X8 ) d[X , X] s in probabil­ ity by Theorem 30. It remains to consider the third sum Ei Xr,n , Xr,f-, . But this sum is majorized, in absolute value, by

R(

)

and since L i (Xr,n+ , - Xr,n ) 2 converges in probability to [X, X]t (Theo­ rem 22), the last term will tend to 0 if limn ---> oo supi r ( IXT,f- 1 - Xr,n I ) = 0. However s f-* X8 (w ) is a continuous function on [0, t], each fixed w, and hence uniformly continuous. Since limn ---> 00 supi ITI.f- 1 = 0 by hypoth­ esis, we have the result. Thus, in the continuous case, f(Xt ) - f(X0 ) = I� f'(Xs-) dX8 + � I� f"(X8_ ) d[X, X8], 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 > 0 we have Eo<s 0 and t > 0, let A = A(c:, t) be a set of jump times of X that has a.s. a finite number of times s, and let B = B(c:, t) be such that Ls EB (�Xs) 2 < c2 , where A and B are disjoint and A U B exhaust the jump times of X on (0, t]. We write

rr l

* * *

,

f(Xt ) - f(Xo ) = I:i { f(Xr,f- 1 ) - f(Xr,n ) } =

L { f (XT,f- 1 ) - f(Xr,n ) } + L { f(Xr,f- , ) - f(Xr,n ) }

""' {f(Xs) - f(Xs_)} , f(Xr1.T>+ 1 ) - f(XrT>1. )} � { sEA i,A

lim ""' � n

i,B

i,A

and by Taylor ' s formula

=

{ f(Xr,f-, ) - f(Xr,n ) } L i,B Li f'(Xr,n )(Xr,f-, - Xr,n ) + � Li f" (Xr,n )(Xr,f-, - Xr,n )2 - I: { !' (Xr,n ) (Xr,f- , - Xr,n ) + � !" (Xr,n ) (Xr;f-, - Xr,n ) 2 } i,A =

+ L R(Xr,n , Xr,+ J·

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 ) dX8 and � f" (Xs-) d[X , X]8• The third sum converges to

I�+

I�+ f' (Xs-

{ s EA

- I: f'(X.-)�X.

+ � f" (X8_ )(�X.) 2 } .

Assume temporarily that J Xs l < k, some constant k, all s < t. Then uniformly continuous, and using the right continuity of X we have

()

c

where r(c+) is lim supo!e: r J . Next let tend to 0. Then to 0, and

f"

is

r(c+ ) [X, X] t tends

{ f(X.) - f(X._) - f'(X.- )�X. - � J"(X._)(�X.)2 } L s EA(e:, t)

tends to the series in ( * * * ), provided this series is absolutely convergent. Let vk inf t > 0 : IXt l > k}, with Xo 0. By first establishing * * * for X l[o,Vk)' 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 lf(y) - f(x) - (y - x)f'(x)l < C(y - x )2 . Then =

(

{

( )

=

I: lf (Xs) - f(Xs-) - f'(X.-)�X. I < C I: (�X.) 2 < C[X, X] t < oo, and Lo<s
Corollary (Ito's Formula) . Let X be a continuous semimartingale and let f be a C2 real function. Then f(X) is again a semimartingale and the following formula holds:

f(Xt ) - f(Xo)

=

+ _! t J" (Xs) d[X, X] d t Xs f'(Xs) 2 lo + lo +

• .

Theorem 32 has a multi-dimensional analog. We omit the proof.

Let X (X 1, . . . , xn ) be an n-tuple of semimartingales, and have continuous second order partial derivatives. Then f(X) is a semimartingale and the following formula holds:

Theorem 33. let f : R.n lR -+

=

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 - o dX8, by

Y8 Y8_ o 8 + �[Y, X]�. t d t _d Xs X 2 lo lo =

The Fisk-Stratonovich integral is often referred to as simply the Stratono­ vich integral. The notation "o" 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) = 1tO+ J'(Xs_) odXs+ Proof

f(Xs) - f(Xs-) - J'(Xs-)�Xs} . { L O<s
Note that f' is C2, so that f'(X) is a semimartingale by Theorem 32 and in the domain of the F-8 integral. By Theorem 32 and the definition, it suffices to establish �[f'(X) , X]c = � J� f"(X8_ ) d[X, X ]�. However

Thus

t J"(Xs-)dXs + �2 Jo+t j<3>(X8_) d[X, X]� j'(Xt) - j'(Xo) = lo+ + L {f'(Xs) - J'(Xs-) - J"(Xs-)�X8 } . O <s< t

[f'(X) , xr = [!"(X-) · X, xr + [ �2 J<3l(X_ ) · [X, X] , xr . The first term on the right side above is J� f"(X8_ ) d[X, X ]� by Theorem 29;

the second term can easily be seen, as a consequence of Theorem 22 and the

7 Ito's Formula (Change of Variables)

83

fact that [X, X] has paths of finite variation, to be Lo<s
Note that if X is a semimartingale with continuous paths, then Theorem 34 reduces to the classical Riemann-Stieltjes formula f(Xt) -f(Xo) J� f' (Xs) dX8• This is, of course, the main attraction of the Fisk-Stratonovich integral. o

=

Corollary (Integration by Parts). Let X, Y be semi martingales, with at least one of X and Y continuous. Then

Proof.

The standard integration by parts formula is

However [X, Y]t [X, Y]f + X0 Y0 if one of X or Y is continuous. Thus adding 0 � [X, Y] f to each integral on the right side yields the result. =

One can extend the stochastic calculus to complex-valued semimartingales.

Let X, Y be continuous semimartingales, let Zt and let f be analytic. Then

Theorem 35.

t

Xt + iyt,

t

f(Zt ) f(Zo) + f'(Zs)dZs + .!_ j" (Zs)d[Z, Z]8• 2 lo+ lo+ =

Proof.

=

Using Ito's formula for the real and imaginary parts of f yields

lo 1

lot

t at (Zs)dYs a t (Zs)dXs + f(Zt) f(Zo) + + O + aY O aX t a2 f t a2 f 1 Y]s (Zs)d[X, X]s + (Zs)d[X, +2 0+ aX2 0+ aXay t a2 f l +2 0+ ay2 (Zs)d[Y, Y] s · Since f is analytic by the Cauchy-Riemann equations U f' and � if' . Differentiating again gives �:{ �{ f" and ::JY if" and �:{ - f". The result now follows by collecting terms and observing that dZ8 dX8 + idYs and [Z, Z] [X + iY, X + iY J [X, X] + 2 i[X, Y] - [Y, Y] . 0 Theorem 35 also has a version for Zt Xt +iYt, with both X and Y cadlag =

1

l

=

=

=

=

=

=

=

=

=

semimartingales. The proof is almost the same as the proof of Theorem 35.

84

II Semimartingales and Stochastic Integrals

Let X, Y be cadlag semimartingales, let Zt = Xt + iyt, and let f be analytic. Then t t f(Zt) = f(Zo) + { j'(Zs - )dZs + �2 J{ J" (Zs - )d[Z, Z]� lo + o+ {f(Zs ) - J(Zs - ) - J'(Zs - )�Z8 }. + O<s
L

zs m

ate to use the non-negative increasing process

[Z, Z] = [X, X] + [Y, Y] to play the role of the quadratic variation. 8 Applications of Ito's Formula

As an application of the change of variables formula, we investigate a simple, yet important and non-trivial, stochastic differential equation. We treat it, of course, in integral form. Theorem 37. Let X be a semimartingale, Xo = 0. Then there exists a

(unique) semimartingale Z that satisfies the equation Zt = 1 + J� Z8_dX8• Z is given by

{

Zt = exp Xt - � [X, X]t

} O<s
where the infinite product converges. Proof. We will not prove the uniqueness here since it is a trivial consequence of the general theory to be established in Chap. V. ( For example, see Theorem 7 of Chap. V.) Note that the formula for Zt is equivalent to the formula

{

}

Zt = exp Xt - � [X, X]f IT (1 + �Xs) exp { -�Xs}. s 1/ 2 on each compact interval ( fixed w ) . Thus it suffices to show

8 Applications of Ito's Formula

85

IT ( 1 + �Xs 1 { 1� X.I
=

so also is exp{log vt} = yt . To show that Z is a solution, we set Kt = Xt - � [X, X]f, and let f(x, y) = yex . Then Zt = f(Kt , St), where St = I1o<s< (1 + �Xs) exp{ -�Xs } · By the - -t change of variables formula we have

[K, S]C = [S, s] c = 0. Note that s is a pure jump process. Hence I�+ eK• - dSs = Eo <s
)

Thus equation ( * simplifies due to cancellation to we have the result.

Zt - 1 = I� Zs_dXs , and

0

Definition. For a semimartingale X, Xo = 0, the stochastic exponential of X, written £(X), is the (unique) semimartingale Z that is a solution of

Zt = 1 + I� Zs_dXs .

The stochastic exponential is also known as the Doleans-Dade expo­ nential. Theorem 37 gives a general formula for £(X). This formula simplifies considerably when X is continuous. Indeed, let X be a continuous semimartin­ gale with X0 = 0. Then

£(X)t = exp { Xt - 21 [X, X]t}.

An important special case is when the semimartingale X is a multiple >. of a standard Brownian motion B = (Bt)t> O · Since >.B has no jumps we have

86

II Semimartingales and Stochastic Integrals

2 A2 A £(AB)t exp{ ABt - 2 [B, B]t} = exp{ ABt - 2 t}. Moreover, since £(AB)t = 1 + A I� £(AB)8_dB8 we see that £(AB)t 2t B, e>. .x2 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 X0 = Y0 = 0 . Then £(X)£(Y) = £(X + Y + [X, Y]) Proof Let Ut = £(X)t and vt = £(Y)t. Then the integration by parts formula gives that Ut vt - 1 = I�+ U8_dV8 + I�+ V8_ dU8 + [U, V]t . Since U and V are =

.

exponentials, this is equivalent to

Letting Wt = Ut Vt, we deduce that Wt = 1 + I� W8_d(X + Y + that W = £(X + Y + [X, Y]) , which was to be shown.

[X, Y]) s

so 0

X be a continuous semimartingale, X0 = 0. Then £(X) - 1 = £(-X + [X, X]) . Proof By Theorem 38, £(X)£(-X + [X, X]) = £(X + (-X + [X, X]) + [X, -X]), since [-X, [X, X]] = 0. However £(0) = 1, and we are done. 0 Corollary. Let

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.

stochastic process X = (Xt)t>O is a standard Brownian motion if and only if it is a continuous local martingale with [X, X]t = t.

Theorem 39 (Levy's Theorem). A

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

+( )

2 t

1 ZsdXs + � 1 t 1 + iu 1 Z8dX8,

Zt = 1 + iu =

t

t 2 Z8dS - � 1 Zsd[X, X]s

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 t0 , 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

{- � (t - s) } . 2

Since this holds for any u E IR we conclude that Xt - X8 is independent of Fs and that it is normally distributed with mean zero and variance (t - s). 0 Therefore X is a Brownian motion. Observe that if M and N are two continuous martingales such that MN is a martingale, then [M, N] = 0 by Corollary 2 of Theorem 27. Therefore if Bt = (Bf, . . . , B;') is an n-dimensional standard Brownian motion, B: Bf is a martingale for i =/=- j, and we have that if i = j, if i =1- 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 =

(X 1 , . . . , x n ) be continuous local martingales such that if i = j, if i =1- j. Then X is a standard n-dimensional Brownian motion.

As another application oflto's formula, we exhibit the relationship between harmonic and subharmonic functions and martingales. Theorem 41. Let X = (X1 , . . . X n ) be an n-dimensional continuous local martingale with values in an open subset D of!Rn . Suppose that [Xi , Xj ] = 0 if 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 \7u(X8) Jo+

•

dX8 + �2

t .6.u(X8) A8 d Jo+

where the "dot" denotes Euclidean inner product, \7 denotes the gradient, and .6. denotes the Laplacian. If u is harmonic (subharmonic) , then .6.u = 0(.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 Levy's characterization of Brownian motion (Theorem 39) allows us to prove a useful change of time result. Theorem 42. Let = (Mt)t>o be a continuous local martingale with =0 and such that limt -+oo , t = oo a.s. Let

[56]).

M

M0

[M M]

[M M] }

T8 = inf {t > 0 : , t > s . Then (Bs, 9s )s >O is a standard Brownian Define 9s = and Bs = motion. Moreover t)t>o are stopping times for Ws)s>o and a.s. O < t < oo. That is, 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 T8 is finite a.s. by the hypothesis that limt -+oo [ t = oo a.s. Therefore the O"-fields 9s = are well-defined. The filtration (98)s>o need not be right continuous, but one can take Hs = 9s + = to obtain one. Note further that { [M, M]t < s = {Ts > t , hence ([M, M]t)t>o are stopping times for the filtration g = (9s)s> O · = s < oo, By Corollary 3 of Theorem 27 we have E{Mf.J = E{[M, M] r. = s identically because is continuous. Thus the time since

:Fr, ([M, M] Mr,.

M

:Fr

M, M] :Fr. +

8

}

}

[ M , M] [ M , changed process is square integrable. Moreover

-

M] rJ

by the Optional Sampling Theorem. Also

E{B� - B;l9s} = E{(Bu - Bs ) 2 l 9s } = E{ ( = E{ = u - s.

Mr -MrJ2 I :FrJ [M, M]ru - [M, M]r.I :Fr,} u

8 Applications of Ito's Formula

89

Therefore BI - 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 B8 = Mrs has continuous paths. However by Corol­ lary 5 of Theorem 27 almost surely all intervals of constancy of [M, MJ are also intervals of constancy of M. It follows easily that B is continuous. It remains to show that Mt = B[M,M]t . Since Bs = Mrs , we have that B[M, M] t = Mr[ M , MJ t , a.s. Since (Ts )s>O is the right continuous inverse of [M, M], we have that T[M, M] t > t, with equality holding if and only if t is a point of right in­ crease of [M, M]. (If (Ts )s>O were continuous, then we would always have that T[M,M]t = t.) However T[M,M] t > t implies that t [M, M]t is constant on the interval (t, T[M,M] t ); thus by Corollary 5 of Theorem 27 we conclude M is constant on (t , T[M,MJ , ) . Therefore B[M,M], = Mr[M , Ml t = Mt a.s., and we are done. 0 f-+

Another application of the change of variables formula is the determination of the distribution of Levy 's stochastic area process. Let Bt = (Xt, yt) be an IR2-valued Brownian motion with (X0 , 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 �R8dN8, where

R8 = VX2s + Y28

and

Ys dXs + Xs dY8• dNs = - Rs Rs

Therefore the integral At = J� R8dN8 = J�(-Y8dXs+XsdYs) 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 Bt = (Xt, yt) be an IR2-valued Brownian motion, B o (0, 0), u E JR. Let At = f� XsdYs - f� YsdXs . Then 1 A . - oo < u < oo. ' ew O < t < oo, } E{ ' cosh (ut ) -

Proof Then

Let

a(t), f3(t) be C 1 functions, and set . a(t) vt = zuAt - 2 (Xt2 + �2 ) + jJ (t). . At - a ' (t) (xt2 + �2 )dt dvt = zud 2 - a(t){XtdXt + ytdyt + dt } + f3' (t)dt = ( -iuyt - a(t)Xt)dXt + (iuXt - a(t)yt)dyt 1 - 2 dt{a'(t)Xf + a'(t) �2 + 2a(t) - 2/]' (t)}.

90

II Semimartingales and Stochastic Integrals

Next observe that, from the above calculation,

d[V, V]t = (-iuyt - a(t)Xt) 2 dt + (iuXt - a(t) Yt) 2 dt = (a(t) 2 - u2 ) (X1 + �2 )dt. Using the change of variables formula and the preceding calculations

dev' = e v, (dvt + 21 d[V, V]t) = ev' (-iuyt - a(t)Xt)dXt + ev' (iuXt - a(t)yt)dyt 1 + 2 ev'dt{ (a(t ) 2 - u2 - a'(t )) (X; + �2 ) + 2f3'(t ) - 2a(t)} . Therefore e v, is a local martingale provided

a' (t) = a(t) 2 - u2 (J'(t) = a(t). Next we fix t 0 > 0 and solve the above ordinary differential equations with a(t0) = (J(to) = 0. The solution is

a(t) = u tanh(u(to - t)) (J(t) = - log cosh (u(to - t)), where tanh and cosh are hyperbolic tangent and hyperbolic cosine, respec­ tively. Note that for 0 < t < t0 ,

Thus ev, , 0 < t < t0 is bounded and is therefore a true martingale, not just a local martingale, by Theorem 51 of Chap. I. Therefore, However, yt0 = iuAt 0 since a(to) = (J(to) = 0. As follows that V0 = - log cosh( ut0 ). We conclude that

Ao

=

Xo

=

Yo

= 0, it

E{ eiuA 'o } = exp{- log cosh (uto) } 1 - cosh( uto) '

and the proof is complete.

0

There are of course other proofs of Levy's stochastic area formula (e.g. , Yor [246] , or Levy [1 50] ). As a corollary to Theorem 43 we obtain the density for the distribution of At ·

8 Applications of Ito's Formula

91

Corollary. Let Bt = (Xt , yt) be an IR2 -valued Brownian motion, B0 = (0, 0), and set At = XsdYs - Y8dX8 • Then the density function for the distri­ bution of At is

J�

J�

1 !A , (x) = 2t cosh(nxj2t) '

- 00

<X<

00 .

Proof

By Theorem 43, the Fourier transform (or characteristic function) of · E { e i u At = At IS } coshI(ut) . Thus we need only to calculate 2I.,. cosh(ut) du. The integrand is of the form f(z) = �i�j = c�::;;t) . Since cosh(zt) has a pole at zo = i2nt , we h ave enx/2t P(zo ) Res(!, zo) = = (zo ) it Q' Next we integrate along the closed curve Cr traversed counter clockwise, and given by y = 0, crl -r < x < r,

J

Cr :

x = r'

y

= t' 1f

x=

- r'

0< -y< -

-t , r > x > -r,

e - iux

cr2 c3

1f

r cr4

-> y > 0, t 1f

-

as shown in Fig. 1 below. y

x= Fig. 1 .

= jt 1f

x=r

-r

The closed curve of integration

Cr = C¢ + c;_ + c;_ + C�.

Therefore fer f(z)dz = 2ni Res (!, zo) . Along c: the integral is

- r e - i( u+ 1 -

";i ) x

cosh( ut)

r j du = e -, �X

e - iUX

du. cosh( ut)

-r r The integrands on C'; and c; are dominated by 2e - r, and therefore ----

92

II Semimartingales and Stochastic Integrals

lim r

r--+oo

}

Cr

f ( z ) dz = (1 + e x n

ft )

-u

e i x du 1 _ 00 COSh ( ut )

= 2ni Res(!, zo )

x/ t e = 2n t

-n

Finally we can conclude

00

2

1 1 00 e -iux 1 du 2n _ 00 cosh(ut) = 2n

(

2ne¥t-

t(1 + e7 ) 1

t( e ;: + e 2�x ) 1 2t cosh( ;� ) ·

) 0

The stochastic area process A shares some of the properties of Brownian motion, as is seen by recalling that At = R8dN8 , where N is a Brownian motion by Levy's Theorem (Theorem 39) , and N and 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 Bt = SUPo<s 0.

J�

-

-

Bibliographic Notes The definition of semimartingale and the treatment of stochastic integration as a Riemann-type limit of sums is in essence new. It has its origins in the fundamental theorem of Bichteler [13, 1 4] , and Dellacherie [42] . The peda­ gogic approach used here was first suggested by Meyer [1 76] , and it was then outlined by Dellacherie [42] . Dellacherie 's outline was further expanded by Lenglart [145] and Protter [201, 202] . These ideas were also present in the work of M. Metivier and J. Pellaumail, and to a lesser extent Kussmaul, al­ though this was not appreciated at the time by us. See for example [160] , [162] , and [ 1 38] . A similar idea was developed by Letta [ 1 48] . A recent treatment is in Bichteler [15] . We will not attempt to give a comprehensive history of stochastic inte­ gration here, but rather just a sketch. The important early work was that of Wiener [228, 229] , and then, of course, Ito [93, 94, 95, 97, 98] . However, it has recently been discovered, by B. Bru and M. Yor, that the young French mathematician W. Doeblin6 had essentially established Ito's formula in early 6

Wolfgang Doeblin (Doblin ) was born in 1915 in Berlin, and after acquiring French citizenship in 1936 was known as Vincent Dobbin. He was killed in June 1940.

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 [1 38] 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 [21 8] ; see also Meyer [1 73] . 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 1 4 is originally due to Lenglart [142] . Theorem 16 is known to be true only in the case of integrands in lL. The local behavior of the integral (Theorems 1 7 and 18) is due to Meyer [171 J (see also McShane [154]) . The a.s.

Exercises for Chapter II

94

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 [134) . That continuous martingales have paths of infinite variation or are con­ stant a.s. was first published by Fisk [73] (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 [21 7] , and it was extended to general semimartingales by Meyer [171 J . Theorem 35 is inspired by the work of Getoor and Sharpe [215] . The stochastic exponential of Theorem 37 is due to Doleans-Dade [49] . It has become extraordinarily important. See, for example, Jacod-Shiryaev [1 10] . 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 [1 13] for related results. The original result is in Levy [150] , and another proof can be found in Yor [246]. 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 Xt = 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 Xn -+ X in ?­ probability, then Xn -+ X in Q-probability.

Exercise 3. Let (X, Y) be standard two dimensional Brownian motion. (That is, X and Y are independent one dimensional standard Brownian motions. ) Let 0 < a < 1 . Set Bt = aXt + V1 a2 yt. -

Show that B is a standard one dimensional Brownian motion. Compute [X, BJ and [Y, B] .

Exercise 4. Let f : IR+

JR+ 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 lL be such that I Ht l

=

1 , all

t. Let Mt = J� H8dB8 • Show that M is also a Brownian motion.

Exercises for Chapter II

95

Exercise 6. Give an example of semimartingales xn , limn --+oo Xf =

t.

Xt

Exercise 7. Let

a.s., all

Y,

t,

but limn--+oo [X n , Y]t =/=-

X, Y such that [X, Y]t a.s., for some

Z be semimartingales, and let Hn , H E

limn --+oo Hn = H in ucp. Let x n = f H:dY8 and limn --+oo [X n , Z] t = [X, Z]t in ucp, all t.

lL .

X = f HsdYs.

Suppose Show that

C2 ,

and that fn and f� converge uniformly on compacts to f and f' , respectively. Let X, Y be continuous semimartingales and show that limn_, 00 [fn (X) , Y] = [f(X) , Y] .

Exercise 8. Suppose f and all fn are

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 = 0. Let X = M + A. Show that [X, X] = [M, M] a.s.

Exercise 10. Let X be a semimartingale and let Q be another probability law, with Q < < P. Let [X, X]P denote the quadratic variation of X considered

P semimartingale. Show that [X, X] Q = [X, X] P , Q-a.s. Exercise 11. Let B be standard Brownian motion, and let T = inf{t > 0 : Bt = 1 or Bt = 2 } (a) Show that T is a stopping time. (b) Let M = Bti\ T and let N = -M. Show that M and N are continuous martingales and that [M, M] t = [N, N] t = t 1\ T. Show that nevertheless M and N have different distributions. Exercise 12. Let X be a semimartinagle such that L:o<s 0. Let f E C 2 and show that L: o<s s < as a

-

.

as well.

X bena semimartingale and suppose Hn and H are in JL. J H8dX8 in Further suppose that H ----> H in ucp. Show that J H'2'dX8

Exercise 13. Let ucp.

---->

be a semimartingale, and let A be a non-negative, continuous, increasing process with A oo = oo a.s. Suppose limt--+oo J�(l + As) -1 dX8 exists and is finite a.s. Show that limt--+oo 1;- = 0 a.s.

* Exercise 14. Let

X

Exercise 15. Let M be a continuous local martingale and assume [M, M]oo oo a.s. Show that limt--+oo [M�k], = 0 a.s.

=

B be a standard Brownian motion and let H be adapted and continuous. Show that for fixed t, Exercise 16. Let

with convergence in probability.

96

Exercises for Chapter II

Exercise 17. Let Y be a continuous, adapted process with Y constant on [1, oo). Let

X;' = n

i�l Y8ds1 {t> ! } · n

Show that xn is a semimartingale, each not be a semimartingale.

Exercise 18. (Continuation of Exercise motion and let

n,

but that

Y0 = 0 and with

limn _,oo xn

=

Y

need

17.) Let B be a standard Brownian

xn = n JL ! Bsds1 {t> ! }· Solve

dZ;' = Z�dX�, Z'(; = 1 , and show that limn ->oo X[' = Bt a.s., each t, but that limn _,oo zn =/=- Z, where Z solves the equation dZt = ZtdBt. Exercise 19. (Related to Exercises 17 and 18.) Let Af = ;. sin( nt) , 0 < t < 1rj2. Show that A n is a semimartingale for each n, and that J02 ldA� I = 1, each n (this means the total variation process, path-by-path, of A), but that sup IAfl = 0. ->oo O
-

-

2

*Exercise 20. (This exercise refers to an example briefly considered in Sect. 6 of Chap. I on local martingles, and again in Sect. 6 of Chap. II.) Let B be

a standard Brownian motion in IR3 , and let u : IR3 \ {0} ----> IR be given by u(y) = IIYII- 1 . Assume Bo = x a.s., where x =/=- 0, show the following. (a) Mt = u(Bt) is a local martingale. (b) limt oo E x (u(Bt)) = 0. (c) u(Bt) E L2, each t > 0. (d) SUPt>o Ex (u(Bt) 2) < 00. ->

u(B) is not a martingale. Exercise 21. Let A, C be two non-decreasing cadlag processes with Ao Co = 0. Assume that both A00 = limt_, 00 At and Coo = limt->oo Ct are finite. (a) Show that AooCoo = j000 (A00 - As)dCs + j000 (Coo - Cs - )dA8• Conclude that

(b) Deduce from part (a) the general formula

Note that the above formulae are not symmetric. See also Theorem 3 Chap. VI.

m

Exercises for Chapter II

Exercise 22. Let A be non-decreasing and continuous, and assume A00 limt->oo At is finite, Ao = 0 . Show that for integer p > 0,

97 =

*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

Q2 = :Ft V a(BI) and gt = nu >t g� . Show that E{Bt - Bs iQs} = � � (B 1 - B8) for 0 < s < t < 1 .

G by

(a) (b) Show that

Bt =

where

t/\1 1 B l - Bs ds f3t +

1-s

0

f3 is a G-Brownian motion. (Note in particular (Bt - J� 8i=�· ds)o
(c) Show that

and conclude that

{1

Bs B l oo }0 I 1 - s Ids < B is a G semimartingale.

a.s.

This is a special case of Theorem 3 in Chap. VI.

Exercise 24. With the notation of Exercise 23, let Xt (a) Show that

(b) Deduce from (a) that

Xt =

t 1 ( 1 - t) 0

that the process

=

Bt - tB 1 , 0 < t < 1 .

1 df3s· 1-s

X satisfies the stochastic differential equation 1

Xs ds . -s

(X is called a Brownian bridge from 0 to 0.) *Exercise 25. Let (Bt, :Ft)t>O be a standard Brownian motion. Let a, b > 0 and T = inf{ t > 0 : Bt E (- a, b)c}. (a) Show that

is a local martingale. (b) Show that

98

Exercises for Chapter II

(c) For 0

<e< � a

b,

show that X T is a positive supermartingale, and deduce

cos(

a + b e)E{ exp{ 1

l 2T}}

2

(d) Use (c) to show that martingale. (e) Conclude that

< cos( a -2 b e) .

XT = sups
cos( ¥ e) 2 { }} e . xp = T � a b E {e 2 cos ( t e)

Exercise 26. Let (Bt, .Ft)t> O be standard Brownian motion, and let T = inf{t > 0 : Bt E (-d, d) c }. Let M = B T . Show that (a) if d

< ; , then E{ exp{ � [M, M] T } } < oo; but (b) if d = ; , then E{expg [M, M] T }} = oo. (Hint: Use Exercise 25.) Exercise 27. Let (Bt, .Ft)t> O be standard Brownian motion, and let Xt = e-<> t (Xo+a J; e "8 dB8 ) . Show that X is a solution to the stochastic differential

equation

dXt = - Xtdt + adBt. o:

Exercise 28. Let B be a standard Brownian motion and let £(B) denote the stochastic exponential of B. Show that limt-> £(B)t = 0 a.s. Exercise 29. Let X be a semimartingale. Show that oo

1

d( £(X) ) Exercise 30. Let

=

-dX + d[X , X] £ (X)

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{eB• }.

The next eight problems involve a topic known as changes of time. For these problems, let ( 0, .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 n, the function R. (w) is non-decreasing, right continuous, Rt < oo a.s., and Ro = 0. Let gt = .FRt . Change of time is discussed further in Sect. 3 of Chap. IV.

Exercise 31. Show that

G = (Qt)t> O satisfies the usual hypotheses.

Exercise 32. Show - that if M is an IF uniformly integrable martingale and Mt := MR. , then M is a G martingale.

99

Exercises for Chapter II

Exercise 33. If M is an IF (right continuous) local martingale, show that

is a

G semimartingale.

M

*Exercise 34. Construct an example where M is an IF local martingale, but

M is not a G local martingale. (Hint: Let (Xn )n EN be an adapted process. It is a local martingale if and only if IXn l dP is a a-finite measure on Fn - 1 , and E{Xn iFn - d = Xn - 1 , each n > 1 . Find Xn where IXn l dP is not a-finite on Fn - 2 , any and let Rn = 2n.) Rs continuous, strictly *Exercise 35. Let R be a time change, with 0, and Rt < oo, each t > 0. Show that for a continuous increasing, R0 n,

s f---+

=

semimartingale X, for bounded

HEL

*Exercise 36. Let

R and X be as in Exercise 35. No longer assume that Rt < oo a.s. , each t > 0, but instead assume that X is a finite variation process. Let At = inf{ > 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, RA, > t, and if R is strictly increasing and continuous then RA, t. (d) Show that for bounded H E IL we have s

=

(e) Show that for bounded H E

IL

we have

1Rt H8dA8 1ti\Aoo HR.ds. =

Ro See in this regard Lebesgue 's change of time formula, given in Theorem 45 of 0

Chap. IV.

R be a change of time and let G be the filtration given by gt = :FR, · Let At = inf { > 0 : R8 > t}. Show that A = (At)t>O is a change of time for the filtration G. Show also that if t --+ Rt is continuous a.s., Ro 0, and Roo oo, then RA, t a.s. , t > 0. *Exercise 38. Let A, G, be as in Exercise 37 and suppose that RA, t a.s., t > 0. Show that QA, C :Ft each t > 0. *Exercise 39. A function is Holder continuous of order o: if lf(x) - f(y)i < Klx -yic.. Show that the paths of a standard Brownian motion are a.s. nowhere locally Holder continuous of order o: for any o: > 1/2. (Hint: Use the fact that lim n oo L 7rn[o ,t] (Bti+ 1 - Bt ; )2 t.) *Exercise 37. Let

s

=

=

=

=

,

->

=

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�which requires 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, oo ) . We write J000 ldA8 I or is the total variation of the paths of A.

I Aioo for the random variable which

Definition. An adapted, cadlag process X is decomposable if there exist processes

N, A such that

102 with

III Semimartingales and Decomposable Processes

No = Ao = 0, N a locally square integrable local martingale, and A an

FV process.

Y is a classical semimartingale if there exist processes N, B with No = Bo = 0 such that Definition. An adapted, cadlag process

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

where

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 j3 > 0, there exist M, A with M0 = A0 = 0, M a local martingale with jumps bounded by /3 , A an FV process, such that Xt = X0 + Mt + At ; (iv) X is a classical semimartingale. p on JR.+

n is the smallest a­ algebra making all processes in IL measurable. We also let P (resp. bP ) denote Definition. The predictable a-algebra

X

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 C lassification of Stopping Times

103

Let X be a semimartingale. If X has a decomposition Xt Xo + Mt + At with M a local martingale and A a predictably measurable FV process, M0 = A0 = 0 , then such a decomposition is unique. Theorem 2.

=

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 > 0}, all and limn _,00 Tn = T a.s. Such a sequence (Tn ) is said to announce T.

n,

If X is a continuous, adapted process with Xo = 0, and T = inf{t : IXtl > c}, for some c > 0, then T is predictable. Indeed, the sequence Tn = inf { t : I Xt I > 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 < oo). 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} = 0.

Let T be a stopping time and A E FT. We define TA (w) =

{

T(w), oo ,

if w E A, if w rf. A.

It is simple to check that since A E FT , TA is a stopping time. Note further that T = min(TA, TAc) = TA 1\ TAc . 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 of R+ 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 Ts is totally inaccessible,

and T = TA 1\ T8 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 R1 = T and take o: 1 = sup{P(S = R1

< oo)

: S is predictable}.

O:i = sup{ P( S = Ri

< oo)

: S is predictable}

Choose S1 predictable such that P(S1 = Rt < oo) > T and set V1 = S1 . Define R2 = R1 {v1 # R d · For the inductive step let If o:i = o we stop the induction. Otherwise, choose Si predictable such that P(Si = Ri < oo) > T and set Vi = Si {s, does not equal any of vj , l <j < i- 1} " Let Ri+l = Ri { V, #R; } . The graphs of the Vi are disjoint by construction (the finite valued parts, of course) . Moreover the sets {Ri+1 =1- Ri , Ri < oo} are disjoint. Thus I: �' < I: P(Ri + 1 =1- Ri, Ri < oo) < 1, which implies that O:i __, 0. The Ri form a non-decreasing sequence, so set U = limi_, oo Ri. If U is not totally inaccessible, there exists a predictable time W such that P(W = U) > 0 and hence P(W = U) > o: 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}. D

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 IF�' = (:Ff)o
Let X be a (strong) Markov Feller process for the probability P�-', where the distribution of Xo is given by p,, and with its natural completed filtration IF�' . Let T be a stopping time with P�-'(T > 0) = 1 . Let A = {w : XT (w) =f. XT- (w) and T(w) < oo } . Then T = TA 1\ TAc , where TA is totally inaccessible and TAo is predictable. Theorem 4 (Meyer's Theorem).

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.

T

be a stopping time. The a-algebra FT- is the smallest a-algebra containing :F0 and all sets of the form A n { t < T}, t > 0 and A E :Ft.

Definition. Let

Observe that FT - C :FT , 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.

Let T be a predictable stopping time and let (Tn ) n > 1 be an announcing sequence for T. Then FT- = a{Un >I :FTn } = Vn >I FTn .

Theorem 5.

3 The Doob-Meyer Decompositions

We begin with a definition. Let N denote the natural numbers.

X is a potential if it is a non­ negative supermartingale such that limt---> oo E{Xt} = 0. A process (Xn ) n E N

Definition. An adapted, cadlag process

is also called a potential if it is a nonnegative supermartingale for N and

limn ---> oo E{Xn } = 0.

A

potential (Xn ) n E N has a decom­ position Xn = Mn - An , where An+ 1 > An a.s., Ao 0, An E Fn - 1 1 and Mn = E{Aooi:Fn } · Such a decomposition is unique. Proof Let Mo = Xo and Ao = 0. Define M1 = Mo + (XI - E{XI/Fo} ) , and AI = Xo - E{XI I:Fo}. Define Mn , An inductively as follows: Mn = Mn- I + (Xn - E{Xn i Fn- d ), An = An-1 + (Xn - 1 - E{Xn i Fn- d ). Note that E{An } = E{Xo} - E{Xn } < E{Xo} < oo, as is easily checked by induction. It is then simple to check that Mn and An so defined satisfy the hypotheses. Next suppose Xn = Nn - Bn is another such representation. Then Mn ­ Nn = An - Bn and in particular MI - NI = AI - B1 E Fo. Thus M1 - NI = E{MI - N1 1Fo} = Mo - No = Xo - Xo = 0, hence M1 = N1. Continuing D inductively shows Mn = Nn , all n. Theorem 6 (Doob Decomposition) .

=

We wish to extend Theorem 6 to continuous time supermartingales. Note the unusual measurability condition that An E Fn- I 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.

(Yn ) n >l be a sequence of random variables converging to 0 in L2 • Then supt E{ Yn iFt} --+ 0 in L2 . Proof Let Mr = E{ Yn 1Ft} which is of course a martingale for each fixed n. Using Doob's maximal quadratic inequality, E{supt (Mr)2} < 4E{(M�Y} = D 4E{Y,;} 0. Definition. We will say a cadlag supermartingale Z with Z0 = 0 is of Class D if the collection { ZT T a finite valued stopping time} is uniformly Lemma. Let

--+

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 Z0 = 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 A0 = 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 = M0 - N0 = 0 - 0 = 0 for all t. Thus M = N and

A = C.

D

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

E{Coo - Ck iFk } < N a.s. for all k. Then E{C!} < 2N2. Proof of Lemma 1. First observe E{Coo} = E{E{Coo - GolFo}} < N. Let­ ting Ck = Ck+ l - Ck > 0, we obtain by rearranging terms: such that

Thus it follows that

107

3 The Doob-Meyer Decompositions

Choose and fix a constant

Dn = { k2 -n 0 < k2 -n < } Lemma 2. Let T be a totally inaccessible stopping time. For 6 > 0, let R(6) = supt< v P(t < T < t + 61Ft) · Then R(6) ----> 0 in probability as 6 ----> 0. Proof of Lemma 2. Let a > 0 and Sn (6) = inft{t E Dn : P(t < T < t + 6lFt) > a} !\ . 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) = 0. Suppose r c {T < t}, and also T E Ft. Then E{E{l{t< T 0. Then for some t E Dn , P(T < t, Sn (6) = t) > 0. Let r = { T < t, Sn (6) = t}. Then from the definition of Sn (6) we v

E Z+ and let

:

v

.

v

obtain

a contradiction. Thus we conclude Sn (6 ) < T a.s. Next we define a stopping time S as follows. Let S(6) = infn Sn (6) and S = supn S(�). Fix n , so that S(�) = infk Sk ( � ) . Thus on { S = T}, S(�) < S. Hence since S = supn S(�), we have S is accessible on { S = T}, which implies that T is accessible on { S = T }, which in turn implies P(S = T) = 0. Consequently E{l {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 0. Let (3 > 0 and 6 < (3. For n sufficiently large

P(E{l{ Sn (8 ) < T<Sn(8 ) +8 } 1Fsn(8) } > a) > c. We also have P(T = S(6)+6) = 0, since if not S(6)+6-,!. could announce part of T which would contradict that T is totally inaccessible. Thus P( { Sn ( 6) < oo where the symbol � T < Sn (6) + 6} � {S(6) < T < S(6) + 6}) ----> 0 as 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 n ---->

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( S( 6) < T < S(6) + f31Fs ( .s) ) > a on a set A, with P(A) > c. From the definition of S and working on the set A, this implies P(E{l{ s < T < S+.B} IFs - } > a) > c. Let 0 (3 ----> 0 to obtain P(T = S I Fs- ) > a, and we have a contradiction.

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(6) = supt
Proof of Lemma 3. Since IZI < N we have IW(6) 1 < 2N. Thus it suffices to show W(6) --. 0 in probability. Let a > 0 and b = avfE. Define zt = 2:s b} ' zt- = 2:s b } , Zf = Zt - (Zi + Zn , E,

and Wb(6) = SUPt
- Z� IFt}l,

with

w+ (6)

and

w- (6)

defined

-

E { sup 1z; - Z� I I Ft}, Wb(6) < sup t r < s < r+8 which by Doob's inequality implies

However since

zb is cadlag and I.6.Zbl < b, Fatou's Lemma implies

Also, again using that Z is cadlag, for each fixed w the number of jumps larger than b is finite. Define inductively

> 0 : .6.Zt > b}, 1i+l = inf{t > 1i : .6.Zt > b}. Since IZI < N be hypothesis, I.6.ZrJ < 2N. Choose k such that P(Tk < v) < T1 = inf { t

E.

Then

By the previous lemma we know that R( 6) --. 0 in L2, so by taking 6 small enough, we get the above expression less than 2e. The reasoning for w- is analogous. We achieve W(6) < Wb(6) + w + (6) + w- (6) , which gives W(6) --. 0 in L 2 . D 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 = i nf { t > 0 : IZtl > N - c} 1\ N, and zr: = Zti\TN " Then IZf:l <

3 The Doob-Meyer Decompositions

109

IZf I + c < N, and ZN is constant after TN . Thus we are now reduced to the case considered by the Lemma 2. Fix n and let :Fkn = :F Define 2"" k

•

ak = E{Z k - 1 - Z k I R:k_ 1 } zn:-

2"f

All the a'k > 0 since Z is a supermartingale, and also ak E FJ:_ 1 . Let Af: 2::;= 1 aj . Then L k = Z 2"n + A'k is an :FJ: discrete time martingale. Define

=

1 k k n n Bt Ak 1"f 2 n < t <- -. 2n We want to show that E{supt IBf - Bf l 2 } --. 0 as n, m --. oo. Suppose m > n. Since Bm and B n are constant on intervals of the form (k2- m , (k + 1)2 - m ], the sup of the difference occurs at some k2 - m . Fix t and let u = inf{s : s E Dn and > t}. Observe that E{A� ­ Ak' I :Fk } = E{Zt - Zooi:Ft } is bounded by 2N. Also E{A � - B�I:Ft} E{E{A� - B� I:Fu }i:Ft} = E{E{Zu - Zooi:Fu }I:Ft} = E{Zu - ZooiFt} which is bounded by 2N. This implies E{B:;:; - B� - ( Bf - Bf) I:Ft } = E{Zt - Zui:Ft} , with the right side being nonnegative and bounded by W(2 -n ), and of course also by E{W( 2 -n ) I:Ft}. Using Lemmas 1 and 2 we get that E{supt IB� Bfl 2 } --. 0 as n, m --. oo. Therefore the sequence Bf is a Cauchy sequence and we denote its limits by At . Next we show that the process A has continuous paths. We do this by analyzing the jumps of the approximating processes B n . Note that for t k2 -n llB� = E{ Z k-1 - Z.2JLn !Rk- d which are bounded by W(T n ). But then supt l ll Bf l --. 0 in L2 . Thus there exists a subsequence (nj ) such that supt I�B:j I --. 0 a.s. , and hence the limit A is continuous. It remains to show that Zt + A t is a uniformly integrable martingale. Both =

s

=

=

'

�

are square integrable, so we only need concern ourselves with the martingale property. Let s, t E Dn , s < t, and A E :F8 • Then E{(Zt + At) 1A} = E{(Zs + A8)1A} · The result follows by taking limits of Zt + Bf. At the beginning of the proof we made the simplifying assumption that general case. Choose N and let Z had bounded jumps. We now treat the T = inf{t > 0 : IZt l > N} 1\ N. Then zr (that is, z stopped at the time T) has at most one jump bigger than 2 N . By localization and the uniqueness of the decomposition, it suffices to prove the result for this case. Thus without

1 10

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: �Zs l{ t�Z. > 2N }

zt- =

s
=

�Zs l{�Z.< - 2N} L s
� Zrl{�ZT > 2N } l {t> T}

=

�Zrl{�ZT < - 2N.} l{ t>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 - -J:: A - t = Z,+ M+ - M- is a supermartingale with jumps bounded by 2N. Let Z = M - A be the unique decomposition of Z which we now know exists. Then Z = Z + M+ - M - , and therefore Z = CM 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�Zr l < IZr - 1 + IZrl < N + IZrl E L 1 , and hence E{ I� Zrl} < oo. Choose 0. Next choose R N so large that E{ I�Zrl l {I .6. ZT I> R} } < 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 = � Zr l{t > T }l{ - .6-ZT > R } , and Zf = zt- - Zf. We define Bn- , BnR , Bnd analogously to the way we defined Bn in equation ( * ) above. We first show Bf - converges uniformly in t in probability. We have:

+

t:a.

P(sup i B� -

a, t: >

>

- B;" - I > a )

(* * )

t < P(sup t I B�d - B;"d l >

a ) + P(sup IB�R I > a ) + P(sup IB;"R I > a ) t t 3 3 3 The second term on the right side of ( * * ) above is less than 3t::

3t:

The third term on the right side of ( ** ) is shown to be less than similarly. Since I ZR 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 Bn . Thus as we did before in this proof, we can conclude that B�+ converges uniformly in t in probability as n ----> oo, and we denote its limit process by Ai.

3 The Doob-Meyer Decompositions

111

We prove continuity of A exactly as before. Finally by taking a subsequence n1 such that Bnj + converges almost surely, we get

+ } < lim inf E{B� + } = -E{Z�,} B� E{A;t,} = E{ njlim ---+ oo n1 ---+ 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 Z0 = 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 A0

=

0 is of integrable variation if the

expected total variation is finite: E{fr�' ldA8 I } < oo. A shorthand notation for this is E{IAioo} < oo. An FV process A is of locally integrable variation if there exists a sequence of stopping times (Tn ) n > 1 increasing to oo a.s. such n r that E{J0 ldAs l} = E{IA irn } < oo, for each n.

Definition. Let

A be an (adapted) FV process, A0 = 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.

Let A be an FV process, Ao = 0, and E{IAioo} < oo. Then A is natural if and only if

Theorem 9.

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�} < oo, hence N is a true martingale (Theorem 51 of Chap. I). Therefore E { J000 As - dMs } = E{ N00 } - E{ No} = 0, since N is a martingale, so that the equality holds if and only if E{[M, A] 00 } = 0. D

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{J0T Hs_dA s } = 0, as is easily seen by approximating sums and the Dominated Convergence Theorem, since J: ldA8 I E L 1 and E{Ar} = 0. Using the naturality of A, E{Hr Ar } = E{f0T H8_dAs } = 0, and letting Ht = E{l{A r>O} I Ft} then shows that P(Ar > 0) = 0. Since D E{ Ar } = 0, we conclude Ar 0 a.s., hence A 0. Theorem 11. Let A be an FV process of integrable variation with A0 = 0 . If A is predictable, then A is natural. Theorem 10.

-

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.) Let M be a bounded martingale. First assume A is bounded. Then the stochastic integral Jr;'' As dMs 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{f000 AsdMs } = E{AoMo} = 0, since Ao = 0. How­ ever E{fr�' A s_dMs } = E{Ao - Mo} = 0 as well, since Ao - = 0. Further,

Proof.

1

00

AsdMs -

1

00

A s_dMs = =

=

00

1 (As - As- )dMs 00 1 LlA8dM8

2: LlA8LlMs O<s
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] = } = lim E{ [M, An ] = } = 0, n 0 by the Dominated Convergence Theorem. Therefore A is natural. 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.

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. 0 Proof. This theorem is a combination of Theorems 10 and 1 1 . Proof of Theorem 8 (Doob-Meyer: General Case). We begin with the unique­ Theorem 12. =

Z

Z

ness. Suppose = M - A and = 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 M0 - No = 0, it is identically zero. This gives the uniqueness. The existence of the decomposition is harder. We begin by defining stop­ is in the half-open interval ping times Tn ,j, where Tn ,j is the j-th time bounded by 2 -n and 2-(n - l ) , where -oo < n < oo. 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 with disjoint graphs, each one of which is either predictable or totally inaccessible, which exhaust the jumps of (in the sense that the jump times of are contained in the union of the graphs of the and are such that for each i there exists a constant bi such that

I � Ztl

(Si ) i > l Z

Z

S ) i b; < I� Zs,l < 2b; . We define Z0 ( t ) Zt and inductively define Zi+ 1 ( t ) Zi ( t ) and A; ( t ) 0 if Si is totally inaccessible; whereas Ai ( t ) E { � Z ,I.Fs }1{ s,
+1

+

=

=

oo

= -

=

s

,-

=

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 )IFs, - } = limn E{�Zi(Si )IFs,.' }. Now fix n: E{�Zi (Si )IFsn } = lim E{Zi (Si ) - Zi (Sf ) IFs,. } = lim E{E {Zi (Si ) - Zi (Sf ) I Fsd } i F s,. ' k < 0, and thus Ai is increasing. To see that Zi+ 1 is a supermartingale, it will suffice to show that whenever ul and u2 are stopping times with ul < u2 then E{Zi+ l(Ul)} > E{Zi+ l (U2 )}. Again letting Sf be an announcing sequence for si we have E{Zi (Ul)} - E{Zi(U2 )} = E{Zi (U1 )} - E{Zi((Ul V Sf) 1\ U2 )} + E{Zi ((Ul V Si) 1\ U2 )} - E{Zi ((Ul V Si ) 1\ U2 ) } + E{Zi ((Ul V Si) 1\ U2 )} - E{Zi (U2 )}

'

'

k

'

where each of t he summands on the right side above are nonnegative. Then let n ----> oo to get E{Zi (U1 )} - E{Zi (U2 )} > 0. Finally we want to show that E{I:; = l Aj (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{Z8 k - Zs } -E{" � �Zs } = lim � k A;

1

1

1

< lif L (E{Zs;vs1_ , } + E{Zs1 _ 1 - Zs;vs1 _. } A,

+E{Zs; - Z00 } + E { Zo - Z8k }) = 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 2::7 1 Ai (t) converges uniformly in t a.s. , as h ----> oo, and we call the limit A = (t). Fatou 's Lemma gives us that A = (t) is integrable. Each A(t) is easily seen to be predictable, and hence A00 is the almost sure limit of predictable measurable processes, so it too is predictable. Next set Z= (t) = Zt + A00 (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 Z00 ( t) is a supermartingale. Moreover since each Zi is cadlag, again using the uniform convergence of the partial sums we obtain

115

3 The Doob-Meyer Decompositions

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{.6.Z00 (T)IFr - } = 0 for all predictable times T. We can then show as before that ,W(J) 0 in probability as J ---> 0 for Z00 , and we obtain Z00 (t) = Mt - A(t). The process At = A 00 (t) + A(t) is then the desired increasing predictable process. D --->

'

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 = Z0 + M - A where M is a local martingale and A is an increasing process which is predictable, and M0 = A0 = 0. Such a decomposition is unique. Moreover if limt _, 00 E{Zt} > -oo, then E{A00 } < oo. Proof. First consider uniqueness. Let Z = Z0 + M - A and Z = Zo + N - C be two decompositions. Then M - N = A - C by subtraction. Hence A - C n is a local martingale. Let M T be a uniformly integrable martingale. Then n

and therefore E{A; } < E{Zo - Zt}, using Theorem 17 of Chap. I. Letting n tend to oo yields E{ At} < E{ Zo - Zt}. Thus A is integrable on [0, to ] , each t0 , as is C. Therefore A - C is of locally integrable variation, predictable, and a local martingale. Since A0 - C0 = 0, by localization and Theorems 10 and 11, A - C = 0. That is, A = C, and hence M = N as well and we have umqueness. in­ Next we turn to existence. Let = inf { t : IZtl > m } 1\ m. Then crease to oo a.s. and since they are bounded stopping times Zr = E L 1 each m (Theorem 17 of Chap. I) . Moreover the stopped process zr= is dominated by the integrable random variable max(IZr= I , m ) . Hence if X = zr= for fixed m , then 1t = { Xr ; T a stopping time } is uniformly integrable. Let us implicitly stop Z at the stopping time Tm and for n > 0 define Y';;n = Zt - E{Zn i Ft} , with Y';;n = Y';;'J,n = 0 when t > n. Then yn is a positive supermartingale of Class D and hence Y';;n = Y0n + Mtn - Af. Letting N!: = E{Zn iFt}, a martingale, we have on [0, n] that

rm

rm

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 Finally, note that since E{At} < E{Zo - Zt}, and since A is increasing, we have by the Monotone Convergence Theorem

rm.

1 16

III Semimartingales and Decomposable Processes

E{Aoo} = tlim -+ oo E{Zo - Zt} -+oo E{At} < tlim which is finite if limt-+oo E{ Zt} >

D

-oo.

4 Quasimartingales

X be a cacllag, adapted process defined on [0, oo]. 1 Definition. A finite tuple of points T = (t0 , t 1 , . . . , tn+ l ) such that 0 = to < t 1 < · · · < tn + l = oo is a partition of [0, oo ] . Definition. Suppose that is a partition of [0, oo] and that Xti E L 1 , each ti E T. Define n C(X, T) = 2: IE{Xti - Xt;+l i FtJ I. Let

T

The variation of X along

i=O is defined to be

T Var7 (X) = E{C(X, T)}.

The variation of X is defined to be

Var (X) = sup Var7 (X), T

where the supremum is taken over all such partitions.

Definition. An adapted, cadHtg process X is a quasimartingale on

if E{IXtl}

< oo, for each t, and if Var(X) < oo.

[0, oo]

Before stating the next theorem we recall the following notational conven­ tion. If X is a random variable, then

x+ = max (X, O) , x - = - min(X, 0).

[0, oo ) , we set Xoo = 0. Theorem 14. Let X be a process indexed by [0, oo ) . 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. Also recall that by convention if X is defined only on

1

It is convenient when discussing quasimartingales to include oo in the index set, thus making it homeomorphic to [0, t] for 0 < t < oo. If a process X is defined

only on

[0, ) we extend it to [0, ] by setting X00 0. oo

oo

=

4 Quasimartingales

117

Proof For given s > 0, let I:(s) denote the set of finite subdivisions of [s, oo] . For each T E I:(s ), set

YST = E{C(X, T) + [ .Fs} and z; = E{C(X, T) - [ .Fs} where C(X, T) + denotes L:t; E E{Xt; - Xt;+1 [ .Ft; }+, and analogously for C (X, T) - . Also let --< denote theT ordering of set containment. Suppose a, T E I: (s) with a --< T. We claim Y8cr < Y87 a.s. To see this let a = (to, . . . , tn ) · It suffices to consider what happens upon adding a subdivision point t before to, after t n , or between ti and ti + l· The first two situations being clear, let us consider the third. Set

A = E{Xt; - Xt [.Ft; } ; B E{Xt - Xt;+1 [.Ft} ; C = E{Xt; - Xt; + 1 [ .FtJ; then C = A + E{B[.Ft; }, hence c + < A+ + E{B[ .Ft; } + < A + + E{B + [.FtJ , =

by Jensen's inequality. Therefore and we conclude Y.cr < Y87 • Since E{Y87} is bounded by Var(X), taking limits in £ 1 along the directed ordered set I: (s) we define

and we can define Z8 analogously. Taking a subdivision with t0 = s and tn + l = oo, we see YST - z; = E{c+ - c- [.J=·s } = Xs , and we deduce that Ys - Z8 = X8• Moreover if s < t it is easily checked that Ys > E{Yt[.Fs} and Z8 > E{Zt[.F8}. Define the right continuous processes yt Yt+ , Zt = Zt + ' with the right limits taken through the rationals. Then Y and Z are positive supermartingales and Ys - Z8 = X8• For the converse, suppose X = Y - Z, where Y and Z are each positive supermartingales. Then for a partition T of [0, t] -

= E{Y0} + E{Z0} - (E{yt} + E{Zt}). Thus X is a quasimartingale on [0, t], each t > 0.

D

118

III Semimartingales and Decomposable Processes

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 = 0. 0 Proof This theorem is a combination of Theorems 13 and 14.

Theorem 1 5 (Rao's Theorem) .

5 Compensators

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 words, there exists a unique, where A is a predictable-FV process. In other predictable FV process A such that A - A is a local martingale. -

0, _with locally inte!Q"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 i� also increasing. We also make the obvious observation that E{At} = E{At} for all t, 0 < t < oo Theorem 16. Let A be an increasing process of integmble variation, and let H E L be such that E{f; H8dA8} < oo Then, Definition. Let

A be an FV process with A0

=

.

.

Since A - A is a martingale, so also is has constant expectation equal to 0.

Proof

J H8d(As - A8 ) , and hence it

0

In Chap. IV we develop stochastic integration for integrands which are predictable, and Theorem 16 extends with H E L replaced with H predictable. One of the simplest examples to consider as an illustration is that of the Poisson process N = (Nt)t>O with parameter .X. Recall that Nt - .Xt is a martingale. Since the process At = .Xt_is continuous and obviously adapted, it is predictable ( natural ) . Therefore Nt = >.t, t > 0. 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 'flt = l{t> T }, where T is a nonnegative random variable. Let lF be the minimal filtration making T a stopping time.

5 Compensators

119

Let P(T = 0) = 0 and P(T > t) > 0, each t > 0. Then the lF compensator A of 7], where 7Jt = l{t>7 } , is given by

Theorem 17.

t r A 1 dF At = J (u) , o 1 - F(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 A t = - ln( l - F (T 1\ t)) . Before proving Theorem 1 7 we formalize an elementary result as a lemma.

Lemma. Let (n, F, P) be a complete probability space. In addition, suppose

a positive F measurable random variable and no = a {T 1\ t } , where o n = n v N, N are the null sets of :F, and :Ft = nu >t :F2. Then lF so constructed is the smallest filtration making T a stopping time. Let Y E L 1 (:F). Then

T is

By the hypotheses on T the a-algebra n is equal to the Borel a­ algebra on [0, t] together with the indivisible atom (t, oo). Observe that n = D a{ T 1\ u; u < t } , and the result follows easily.

Proof.

Proof of Theorem 1 7. Fix to > 0 and let be a sequence of partitions of [0, to] with liiDn-Hx' mesh (nn ) = 0. Define Ar = L:1r n E{ 7Jt;+' - 7Jt; 1Ft; } for 0 < t < to. Then 'lrn

} } { E 77t;+, >t; l{T { F. } 1 E 77t;+I I t ; = 7Jti+l {T t l {T>t;}

i)

by the lemma preceding this proof. The first term on the right above is 7Jt;+, l{t;>7} = l{t;+1 >7}l {t;>7} = l {t;>T} · Furthermore, l{7>t;}7Jt;+' = 1 {T > t;} l{t;+1>T } = l{t; <7
Therefore,

These simple calculations yield

120

III Semimartingales and Decomposable Processes

which are Riemann sums converging to J�M (1 maining claims of the theorem are obvious.

-

F(u-) ) -1 dF(u).

The re­

O

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 -¢ (x) . We have that At = - ln ( 1 - F(T 1\ t)) and equivalently we can write At = ¢( T 1\ t). In the case of this corollary, the 0 function cp (x) = x, and the result follows.

Remark. Theorem

17

gives an explicit formula for the compensator of a process 'T}t = l {t> 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 L a G stopping time, and then calculate the G compensator of J.Lt = 1{t>L } ? This has a positive answer with the right hypotheses. To see how to do this, let Xt = 1 (O , L 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 Mt = E{ AooiFt}.) Using the techniques presented in Chap. VI, one can show that if is "the end of an IF predictable set," then the G compensator of J.Lt = 1{t>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 , oo ] x n such that L (w) = sup{t < oo : (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.

L

Theorem 17 treats the case of a point process with one jump. A special case of interest is counting processes. We have the following.

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 Theorem 18.

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 oo a.s. Moreover I Nu,Tn I = Nti\Tn < n and thus N is locally of 0

bounded variation, hence certainly locally of integrable variation.

Nt = Ei > 1 l {t> T;} be a counting process without explosions and let Si = Ti - Ti-1 be its interarrival times, where we take T0 = 0. Let J1 = a{Ns ; s < t}, and let :Ft = J1 v N, the filtration completed by the :F-null Let

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 P(S > x) rather than the customary P(S < x)):

F1 (t) = P(S 1 > t) Fi (s 1 , . . . ' Si- 1 ; t) = P(Si > t 1 S1 = s 1 , . . . ' si - 1 = Si- 1 ) where the s1 are in [0, oo] , j > 1 . (If one of the s1 takes the value +oo, then

the corresponding cumulative distribution function is concentrated at +oo.) Define

t 1 ¢i (s 1 , . . . , si - 1 ; t) =

Fi ( s 1 , . . . , si-1 ., s-) dFi (s 1 , . . . , s i - ; s) . describe the compensator of N for its minimal -1

1

o

We now have the tools to completed filtration. We omit the proof, since it is simply a more notationally cumbersome version of the proof of Theorem 17.

Let N be a counting process without explosions and let lF be its minimal completed filtration. Then the compensator A of N is given by At = ¢ 1 (Sl ) + c/>2 (81 ; S2 ) + · · · + c/>i -1 (81 , . . . , Si - 2 ; Si - 1 ) + ¢i (S1 , . . . , Si- 1 ; 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 ¢i defined in equation ( * ) have the Theorem 19.

simplified form

and the compensator

A is given by i-1

At = 2::.: [2::.: ¢J ( SJ ) + cpi (t - Ti )]l{ T; 1 j = 1 Example (hazard rates and censored data) . In applications, an intuitive interpretation often used is that of hazard rates. Let

T be a positive random

122

III Semimartingales and Decomposable Processes

variable with a continuous density for its distribution function F, given by The hazard rate A is defined, when it exists, to be

f.

1

lim P(t < T < t + h i T > t). A(t) = h--+0 h 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 > t) = Ah + o(h). By Theorem 1 7 we have that the compensator of Nt = l{t> T } is At = J�I\T A(s)ds. Or in language closer to stochastic integration,

Nt -

1

ti\ T

A(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 {t> X }l{ u> T} ' and Nf = l {t>x}l{ T > U } with Ft = a { Nu , N;j; u < t}. Let A be the hazard rate function for T. A is known as the net hazard rate. We define A#, known as the crude hazard rate, by

A# = �� � P(t < T < t + hiT > t, U > t)

when the limit exists. We then have that the compensator of N is given by At = f� l{x>u} A#(u)du = J�I\X A#(u)du = J�I\TI\ U A#(u)du, or equivalently

t

Nt - lo l{x> u } A# (u)du If we impose the condition that as

is a local martingale.

A = A#, which can be intuitively interpreted

P(s < T < s + dsiT > s) = P(s < T < s + dsiT > s, U > s),

then we have the satisfying result that

t

Nt - lo l{x> u } A(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)t>o, 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 > 0. 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 Y are 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

(X, Y) =

2

1 ((X

+ Y, X + Y)

- (X, X) - ( Y, Y) ) .

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 oo a.s. such that n T E{f0 l d [X, Y] s l } < oo for each n. Also, (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, YJ , 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).

Let A be an increasing process of locally integrable variation, all of who�e 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. 0 Proof. Both theorems are simple consequences of Theorem 7. Corollary. Let A be an increasing predictable process of locally integrable variation and let T be a stopping time. If P(Ar =f. Ar - ) > 0 then T is not Theorem 20.

totally inaccessible.

124 III Semimartingales and Decomposable Processes Proof _§uppose 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, Mr -/=- Mr- on {T < oo } ). Proof Define Ut l{t> T} · -

=

=

Then U � an increasing, bounded process of integrable variation, and we let A = U 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.

Let T be a stopping time and let H be an integrable random variable such that E{H IFr - } = 0 on {T < oo}. Then the right continuous martingale Ht = E{HIFt} is zero on [O, T) = {( t , w) 0 < t < T (w) } . Proof Since the martingale (Ht)t>o is right continuous it suffices to show that Htl{t < T} = 0 almost surely for all t. Let A E Ft. Then A n {t < T} belongs both to Fr - and also to Ft. Hence 0 = E{H lAn {t< T } } = E{Htl An {t< T } } · Since A E Ft is arbitrary, this implies that Ht l{t < T } = 0 a.s. If H is Fr measurable then Hl { T
:

0

theorem is proved.

Let T be a predictable stopping time and let A be increasing, predictable, and locally of integrable variation. Then Ar and �Ar are each Fr - measurable. Proof Let Sn be a sequence of stopping times announcing T. Then Asn E Fsn C Fr - for each n. Since Ar = limn__.00 Asn we have that Ar - E Fr- · To show that Ar E Fr - it suffices to show E{A r H} = 0 for every bounded random variable H such that E{H IFr- } = 0. Let H be such a random variable and let Ht = E { H I Ft}, the right continuous martingale with terminal value H. Then Ht (w) = 0 for 0 < t < T(w) by Le Jan's Theorem (Theorem 23). This implies that the left continuous version of H, Ht- , equals 0 on [0, T]. Then since the hypotheses imply that A is natural, we have 00 E{HAr} = E{ Hs - l{s< T}dAs} + E{[H, A]r} = 0 + 0 = 0. 0 Theorem 24.

-

1

6 The Fundamental Theorem of Local Martingales

125

Theorem 25 (Fundamental Theorem of Local Martingales) . Let M be a local martingale and let {3 > 0. 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 =

I�Msll{I � M. I> ,B } , L O<s
C is locally integrable. By stopping we can assume without loss that M is a uniformly integrable martingale and that Mo = 0. For each Ct(w) is a finite sum. Define

and we want to show w,

Rn = inf{t : Ct > n or IMtl > n}.

I�MRn l < I MRn l + IMRn- 1 < IMRn l + n . Moreover eRn < eRn- + I�MRn I < IMRn I + 2n, which is in L 1 . Since Rn increases to oo, we have C is locally integrable. By stopping we further assume that E{ C00} < oo .

Then

Next we define

At = L �Msl{I � M.I > ,B} , O<s 0}, then P(A) > 0. Let R = TA. By Theorem 3 we can decompose R as R = RA 1\ R B , where RA is accessible and R8 is totally inaccessible. Then P(R B < oo) = 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 I L1Nrk I < 2{3, for each Tk. Thus without loss of generality we can take R = T to be predic!_able. By convention, we set L1Nr = 0 on {T = oo } Since we know that L1A r E Fr - by Theorem 24, we have .

L1Nr = L1Nr - E{L1Nri Fr - } M - A)riFr - } - E{ L1AriFr - } = L1(M - A)r + L1Ar - E{ L1( = L1(M - A)r - E{L1(M - A) r iFr - }. Since IL1(M - A)rl < {3, the result follows. D If the jumps of a local martingale M with Mo 0 are bounded by constant {3, then M itself is locally bounded. Let Tn = inf{ t : IMtl > n}. Then IMu\ Tn I < + {3. Therefore M is a fortiori locally square integrable.

a

n

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 ( 0 , :F, P) be complete probability space and let X be a random variable such that X E £1 , but X f'j_ L 2 . Define the filtration

Pi

=

{:F,{0, n},

0 < t < 1, t > 1,

where :Ft = Pi V N, with N all the P-null sets of F. Let Mt = E{XI:Ft}, 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.

Y be a symmetric random variable with a continuous distri­ bution and such that E{ IY I} = oo Let Xt = Yl{t> l }, and define Example. Let

a{IYI}, {a{Y}, .

g� where

=

0 < t < 1, t > 1,

G = Wt)t>o is the completed filtration. Define stopping times Tn by

{

rn = 0,

oo ,

if IY I > n , otherwise.

7 Classical Semimartingales

1 27

Then rn 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.

A classical semimartingale is a semimartingale. Proof. Let X be a classical semimartingale. Then Xt = 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 26.

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 c adtag quasimartingale is a semimartingale.

D

Proof. By Theorem 15 a quasimartingale is a classical semimartingale. Hence it is a semimartingale by Theorem 2 6 .

Theorem 28.

Proof.

A cadlag supermartingale is a semimartingale.

0

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 x t is a semimartingale. However for a partition T of [0, t],

t; ET t; ET = E{Xo} - E{Xt} ·

Therefore x t is a quasimartingale, hence a semimartingale by Theorem 27.

Corollary. A submartingale is a semimartingale.

0

128

III Semimartingales and Decomposable Processes

We saw in Chap. II that if is a locally square integrable local martingale and H E IL, then the stochastic integral H · 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.

X

X

Let M be a local martingale and let H E IL. 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 Theorem 29.

T = inf{ t >

t

0 : lo ldAsl > m}.

Then E{ft\T ldAs l } < m + (3 + E{I�MT I} < oo , and thus by stopping A can be assumed to be of integrable variation. Also by replacing H by H81{ 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 = 0. 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 O"n be a sequence of random partitions of [0, t] tending to the identity. + 1 ATt ) tends to (H · A in ucp, where a is the sequence Then I: HTn ( ATt n 0 = T0 < Tf < · · < Tt < · · · . Let (nk ) be a subsequence such that the sums converge uniformly a.s. on [0, t]. Then ·

-

•

)t

·

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. D

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. = X0 + M + A be a be a classical semimartingale, and let Let decomposition where M0 = A0 = 0, M is a local martingale, and A is an FV

X

Xt

t t

7 Classical Semimartingales process. Then if the space (!1, :F, (:Ft)t>o, P) supports a Poisson process we can write

1 29

N,

Xt Xo + {Mt + Nt - t} + {At - Nt + t} =

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. be a semimartingale. If X has a decomposition Xt = Xo + Mt + At with Mo = Ao = 0, M a local martingale, A an FV process, and with A predictable, then X is said to be a special semimartingale.

Definition. Let

X

= 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 A0 = 0 and A predictable is called the To simplify notation we henceforth assume X0

canonical decomposition.

Theorem 1 5 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.

Let X be a classical semimartingale with bounded jumps. Then X is a special semimartingale. Proof. Let Xt = Xo + Mt + At be a decomposition of X with Mo = Ao = 0, M a local martingale, and A an FV process. By Theorem 25 we can then also

Theorem 31.

write

Xt = 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 1 5 ) . Therefore �

Xt = Xo + {Nt + Lt} + Bt is the canonical decomposition of X and hence X is special. �

0

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\ Ts , where TA

is accessible and Ts is totally inaccessible. By Theorem 21 it follows that P( l flATB I > 0) = 0. 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{flMT I:FT- } = 0,

M, and hence A, are continuous, using Theorem 7 of Chap. I. Theorem 31 can be strengthened, as the next two theorems show.

and

criteria given in these theorems are quite useful.

D

The

Let X be a semimartingale. X is special if and only if the process Jt = SUP s
Before proving the theorems, we need a preliminary result, which is inter­ esting in its own right.

Let M be a local martingale and let Mt = sups n } . Since MTn is uniformly integrable, and Sn < Tn , we have that Msn is integrable. But we also have that Msn < n V IMsn I which D is in L 1 . Since Sn increases to oo a.s., the proof is complete. Theorem 34.

n.

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 +I� ldAs 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 l = IXt - Xt - 1 < 21Xt l , 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 l < I�Ms l + I�Xs l < 2M; + 18 , 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 : I� I dAs I > n}, and note that I{n ldAsl = IoTn - l dAsl + I�ATn l < n + SUPs< Tn I�As l , which is in L 1 as soon as sup s
A class of examples of special semimartingales is Levy processes with

bounded jumps. By Theorem 31 we need only show that a Levy process Z with bounded jumps is a classical semimartingale. This is merely Theorem 40 of Chap. I. For examples of semimartingales which are not special, we need only to construct semimartingales with finite variation terms that are not locally integrable. For example, let X be a compound Poisson process with non­ integrable, nonnegative jumps. Or, more generally, let X be a Levy process 00 with Levy measure v such that I xv(dx) = oo and/or J � lxlv(dx) = oo . 1 8 Girsanov's Theorem

Let X be a semimartingale on a space (!1, .F, IF, P) satisfying the usual hy­ potheses. We saw in Chap. II that if Q is another probability law on (!1, .F), and Q « P, then X is a Q semimartingale as well. If X is a classical semi­ martingale (or equivalently, if X is decomposable) and has a decomposition X = M + A, M a local martingale and A an FV process, then it is often useful to be able to calculate the analogous decomposition X = N + B, if it exists, under Q . This is rather tricky unless we make a simplifying assumption which usually holds in practice: that both Q « P and P « Q.

Definition. Two probability laws P, Q on (n, F) are said to be equivalent

Q and Q « P. (Recall that P « Q denotes that P is absolutely continuous with respect to Q.) We write Q ""' P to denote equivalence.

if P

«

132

III Semimartingales and Decomposable Processes

in L 1 (dP) such that P, then there exists a random variable = and = 1, where Ep denotes expectation with respect to the law P. We let If

Q

� Z

�

Z

Ep{Z}

Zt = E { ��1 Ft } 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 �� = ( �� ) - l . P

We begin with a simple lemma, the easy proof of which we leave as exercise for the reader.

an

P, and

Zt = Ep {�1 Ft }. An adapted, cadlag process M is a Q local martingale if and only if MZ is a P local martingale. Lemma. Let Q "'

Proof.

See Exercise

D

20.

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 t Lt = Mt - lo ; d [Z, M]s 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 Theorem 35 (Girsanov-Meyer Theorem) .

martingale. We need to show it is a classical semimartingale, with the above decomposition being valid. and are P local martingales, they are semimartingales ( corol­ Since lary of Theorem 26) and

M

Z

J Z_dM + J M_dZ

is a local martingale as well (Theorem 29). Using integration by parts we have

ZM - [Z,M] = f Z_dM + f M_dZ is also a P local martingale. Since Z is a version of Ep{ �j !Ft }, we have }; is a cadlag version of EQ {��1 Ft}; therefore }; is a Q semimartingale. Since P � Q, it is also a P semimartingale. Multiplying equation ( * ) by }; 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(under - ( i )[Z, M] is a Q local martingale. We next use integration by parts Q) : ( � ) [Z, M] = J ; d[Z, M] + J [Z,M] _ d (�) + [[Z,M] , �]. Let N = f [Z, M]_ d (i). Since i is a Q local martingale, so also is N (The­ orem 29). The above becomes: ( 1 ) [Z, M]t = Jort 1 - d[Z, M]s + Nt + [[Z, M], z1 lt Zs Zt 1 = ro z _ d[Z,M]s + Nt + L Ll ( � ) il[Z,M]s J s s t O<s< t1 1 = z- d[Z, M]s + Nt . 0 8

Adding our two Q local martingales yields

N +M-

( � ) [Z,M] = N + M - J �d[Z,M] - N = M - I � d [Z,M]

which, being the sum of two Q local martingales, is itself one. This establishes 0 the theorem. The Girsanov-Meyer Theorem has another version using the sharp bracket . It transforms a local martingale into a Q special semimartingale and gives its canonical decomposition. However an extra integrability hypothesis is needed for the theorem to be true. Note that in the predictable version the integrand is is replaced by its left continuous version z!_ .

P

Let X be a P local martingale with Xo = 0. Let Q be another probability equivalent to P and let Zt = E{�IFt}· If {X,Z) exists for the P probability, then the canonical Q decomposition of X is 1 Xt = (Xt - Jot Zs - d{X, Z)s) + Jto Z1s- d(X, Z)s· Proof. Using the same ideas as in the proof of Theorem 35, we want to find a predictable finite variation term A such that Y = Z(X - A) is a P local martingale. Using integration by parts yields t t/' ZtXt = 1 Zs_dXs + Jo Xs _dZs + [Z, X] = Nt + [Z, X ]t - {X, Z)t + {X, Z)t = Mt + {X, Z)t Theorem 36 (Girsanov-Meyer Theorem: Predictable Version) .

134

III Semimartingales and Decomposable Processes

= I� Z8_dXs � X8_dZ8 Z) M = [Z, X] - (X, Z) P ZX - (X, A = (X, Z), A A T A = I A l n } { t n t ATn - = I 1 [o,Tn ) (s)dA8• ATn - . A A = I A I n,

where both N +I and N+ are P local martingales. Therefore is a local martingale. which of course is a predictable FV process. Moreover Let we know that does not jump at totally inaccessible times, and that we can cover its jumps with predictable times. This implies that is locally > inf > 0 : bounded. Let Tn , and let be defined by From now on we assume Then is still < and the stopping times Tn an FV process and is still predictable, increase to oo a.s. Integration by parts gives us also

[A, i]i ] = EQ{[A, T } i

· If we show that is also a Q local martingale, we will be done. It suffices to show that 0 for every stopping time T. Since is FV and predictable, by stopping at T- if necessary, we can assume the jumps of are bounded. And since is a Q local martingale, its supremum process is locally integrable (Theorem 34). This is enough to give us that is Q locally of integrable variation. By stopping we assume it is integrable. We then have that is Q natural, so if T is a stopping time, we replace with and we still have a natural process, and thus we have 0. Since T was arbitrary, this implies that is a martingale. Of course, the process is implicitly stopped at several stopping times, so what we have actually shown is that is a Q local martingale. We now have that

A

[A, i]

[A, i]

A

[A, i] A AT EQ{[A, i ]T } =

A

1 At = Q-local martingale + 1t 1 dAs t o Zs which in turn gives us, recalling that A = (X, Z), Xt = Z1t (ZtXt ) = Z1t (Mt + (X, Z) t ) t 1 1 1 d(X, Z)s · = -M + Q-local martingale + t Zt 0 sSince Z ( i M) = M = P local martingale, we have that i M is also a Q lo cal martingale. Thus finally X - I� z!_ d(X, Z) s is a Q local martingale, and the t z

z

proof is done.

D

Let us next consider the interesting case where Q is absolutely continuous with respect to but the two probability measures are not equivalent. (That is, Q.) We begin with a simple result, and we will assume a.s.

P, we no longer assume that P is also absolutely continuous with respect to :F0 = { 0,!1} Theorem 37. Let X be a P local martingale with X0 = 0. Let Q be another probability absolutely continuous with respect to P, and let Zt = E { � I :Ft } .

8

13 5

Girsanov's Theorem

Assume that {X, Z) exists for P. Then At = J; z!_ d {X, Z)s exists a.s. for the probability Q, and Xt - J; z!_ d{X, Z) s is a Q local martingale. Proof. Zo = E{Z} = 1 , so if we let Rn = inf{t > 0 : Zt < 1 /n} , then Rn increase to oo , Q-a.s., and the process z:_ is bounded on [0, Rn ] · By Theorem 36, x[ln - J; Z�- d{X, z):-n is a Q local martingale, each n. Since a D

local, local martingale is a local martingale, we are done.

We now turn to the general case, where we no longer assume the existence of {X, Z), calculated with (As before, we take X to be a local martingale. ) We begin by defining a key stopping time: R = inf{ > 0 : Zt = 0, Zt - > 0 }. Note that Q ( R < oo ) = 0, but it is entirely possible that R < oo ) > 0. We further define Ut = �XR 1{t> R} · Then U is an FV process, and moreover U is locally integrable Let Tn increase to oo and be such that XTn is a uniformly integrable martingale. Then

P.

P t P(

(dP).

Thus U has a compensator local martingale.

U, and of course U is predictable and U - U is a �

�

�

P Theorem 38 (Lenglart-Girsanov Theorem) . Let X be a P local martin­ gale with Xo = 0. Let Q be a probability absolutely continuous with respect to P, and let Zt = Ep{ � IFt}, R = inf{t > 0 : Zt = 0, Zt- > 0}, and Ut = �XR I{t> R} · Then t 1 1 Xt - -d[X, Z]s + Ut 0 Zs

is a Q local martingale. Proof. Let Rn = inf {t > 0 : Zt < �}. (Recall thatRn Zo = 1R, nand also note that it is possible that Rn = R . ) Then both X and Z are P local n R Rn Rn martingales. Also note that A� = J; zfn 1 {z:Zn >O} d[X , z ]s , y Rn = x Rn - A Rn + URn are all ?-well-defined. We can define

At =

u n , and

lo �s l{z. >o} d[X, Z]s t

on [O , R) , since d[X Rn , zRn ]s does not charge (R, oo), and z.An 1 { z Rn > o} = 0 at R. Thus we need only to show Y Rn is a Q local martingale for each fixed which is the same as showing that zRn y Rn is a p local martingale. Let us assume all these processes are stopped at Rn to simplify notation. We have 8

n,

ZY = ZX - ZA + ZU. �

136

III

Semi martingales 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-U is a local martingale ( dP). Summarizing we have

ZY = ZX - ZA + ZU = local martingale + [Z, X] - local martingale + l{z>o} d[X, Z]) + local martingale + Z�dU �

J

which we want to be a local martingale under case if

J

dP. This will certainly be the

d[Z, X] - l { z > o}d[X, Z] + Z�dU = 0.

However ( *) equals

�ZR �XR l{tA Rn > R} + ZR� �XR l{tA Rn > R} · But �ZR = ZR - ZR � = 0 - ZR� = -ZR� , and this implies that equation (*)

D

is indeed zero, and thus the Lenglart-Girsanov Theorem holds.

Corollary. Let X be a continuous local martingale under P. Let Q be abso­

(X, Z) = [Z, X] = [Z, X] c exists, t f Xt - Jo s� d[Z, X]� = Xt - Jo asd[X , X]s

lutely continuous with respect to and

P.

Then

t zl

which is a Q local martingale. Proof By the Kunita-Watanabe inequality we have

which shows that it is absolutely continuous with respect to whence the result.

d[X, X]s

a.s. ,

D

8

137

Girsanov's Theorem

is the solution of a stochastic exponential equation Z dZ8 Z8_H8dX8 (which it often is) , then = H8 • 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 dS8 = h(s, S8)dBs + b(s; Sr ; 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{Z I :Ft }, which is clearly a cadlag martingale. By Girsanov's Theorem rJot h(s, Ss)dBs - ro t 1 d[Z, Jro h(r, Sr )dBr ]s J Zs is a Q local martingale. We want to find the martingale Z. In Chap. IV We remark that if of the form =

0! 8

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 = for some predictable process If we assume is well behaved enough to define = t, then which gives us a linear stochastic differential = we have equation to solve for Thus if we let = = we get that It remains to determine We do this by observing from our previous Girsanov calculation that

J. Z Zt 1 + J� H8Z8dB8, Z. H.

Zt 1 + J� J8dB8

H8

Nt J� H8dB8,

Zt

t:(N) t .4 t{ h(s, S8)dBs t ..l_zsHsh(s, Ss)ds = {o t h s, S8)dBs - {o t Hsh(s, Ss)ds - Jo Zs J ( J Jo -b(s; Sr ; <- s) , wh" h yw. lds . is a Q ocal martmgale. vve then choose H8 = h(s, Ss ) St = fo\(s, Ss)dBs + fo\(s; Sr ;r < s)ds t . < s) · b( s r is a local martingale under Q. Letting Mt = Bt + Jo �(:: :; ds denote this Q local martingale, we get that [M, M] t = [ B , B] t = t, and by Levy's Theorem M is a Q-Brownian motion. Finally, under Q we have that S satisfies the stochastic differential equation 1

ur

r

IC

s

s r

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

a priori

There is one problem with the preceding example: we do not know whether our solution Z is the Radon-Nikodym density for simply a measure probability measure. This is a constant Q, or whether Q is an actual problem. Put more formally, we wish to address this problem:

bona fide

Let

M be a local martingale. When is t: (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 is standard = ll x ll - 1 and = where is a uniformly integrable local three dimensional Brownian motion. Then martingale but not a martingale. Nevertheless, whenever is a continuous local martingale, then is a positive supermartingale, as is the case with any nonnegative local martingale. Since = 1, and since M is a positive supermartingale, we have < 1 , all S ee the lemma below.) It is easy to see that Z We begin with some = 1, preliminary results.

t: (

NNt u(Bt )

u(x) M

B

t: (M)

) t:( £(M)0 E{t: t. is a (M)t} ( true martingale if one also has E{t:(M)t} for all t. Lemma. Let M be a continuous local martingale with M0 = 0. Then E{t:(M) t } < 1 for all t > 0. Recall that £(M)0 = 1. Since M is a local martingale, t:(M) is a Proof. nonnegative local martingale. Let T be a sequence of stopping times reducing t:(M). Then E{t:(M)tATn} = 1 , and using Fatou's Lemma, E{£(M)t} = E{liminf £(M)tATn} < lim inf E{£(M)tATn} = 1. Theorem 39 . Let M be a continuous local martingale. Then E {e ! M• } < E{e ! [M, M]t } I /2. Proof (£(M)) ! = ( eM, - ! [M ,M]t ) ]! = e ! M' (e -! [M , M t ) ! n

n--+oo

n--+oo

0

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�M, = (£(M) t )� (e� [M,M],)� and this together with the Cauchy-Schwarz inequality and the fact that E{£(M) t } < 1 gives the result. 0

Lemma. Let M be a continuous local martingale. Let 1

< p < oo , 1 + 1 = 1.

Taking the supremum below over all bounded stopping times, assu�e tbat )

sup E{ec 2J,? Mr } < oo. 1

T

Lq bounded martingale. Proof Let 1 < p < and r = ��� . Then s = �+l and ; + ! = 1 . Also we note that (q - J"!)s = j/ which we use in the last equality of the proof. Then £(M) is an

oo

We have

2(

l)

£ (M) = eqM- 1 [M,M] = evrM- 1 [M,M] e(q- vr) M We now apply Holder ' s inequality for a stopping time

S:

E{£(M) �} = E{ e vqrMs - y [M, M]s } � E{ e(q-vr) Ms } � = (E {£( y'q"TM) s } ) r ( E{e ( 2<0'0' 1 ) ) Ms }F 1

1

E{£(y'qrM)s} < 1, we have the result. M Theorem 40 (Kazamaki's Criterion) . supr E{ e ! M ) } < oo, £ ( M) Recalling that

0

Let be a continuous local mar­ tingale. Suppose over all where the supremum is taken e bounded stopping times. Then is a uniformly integrable martingale. Proof. Let 0 < a < 1 , and p > 1 be such that (Jf!-l) < ! . Our qhypothesis combined with the preceding lemma imply that £(aM ) is an L bounded ! = 1, r

martingale, where + ! martingale. However

which in turn implies it is a uniformly integrable

£(aM) = ea M- 4 [M, M] = ea2 M- a22 [ M, M] ea ( l - a) M £(M t2 ea(l - a) M , =

2 and using Holder's inequality with a - 2 and ( 1 - ) - 1 yields (where the 1 on the left side comes from the uniform integrability) :

a

140

III

Semimartingales and Decomposable Processes

a

1

Now let increase to and the second term on the right side of the last < ___, 0. Thus inequality above converges to since 2 ( and since we know that it is always true that > E{ ( ) } , we are done. 0

a 1 a) £ M 1 E{£(M) 00 }, 1 00

1

-

As a corollary we get the very useful Novikov's criterion. Because of its importance, we call it a theorem.

Theorem 41 (Novikov's Criterion) .

Let M be a continuous local martin­

gale, and suppose that E{e ! [M,M]oo } Then £ (M) is a uniformly integrable martingale. Proof By Theorem 39 we have E{e! Mr } E{d(M,Mlr } ! , and we need only to apply Kazamaki's criterion (Theorem 40). 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, N]t = J� H';ds where H = h(s�Ss) . Clearly whether or not 1 [N, N] t has a finite exponential 2 moment will depend on our choice of b and of h. A standard simple solution is to choose b and h so that b / h is bounded. Partial results exist that give an analogue of Novikov's criterion in the general (cadlag) case. See for example Exercise 14 of Chap. V, and the sources [156] and [157]. Example. This example is taken from statistical communication theory . Sup­ pose we are receiving a signal corrupted by noise, and we wish to determine <

00 .

<

0

b(s · S - r <s)

if there is indeed a signal, or if we are just receiving noise (e.g., we could be searching for signs of intelligent life in our galaxy with a radio telescope) . Let the noise, and the actual (transmitted) be the received signal, signal. Then

x(t)

s(t) e(t) x(t) = s(t) + e(t ).

A frequent assumption is that the noise is "white." A white noise is usually described as a second order wide sense stationary process with a constant spectral density function (that is, E{�t } = 0, and = E{ XtXt+r } = 8r So , where Or is the Dirac delta function at T , and S0 is the constant spectral Such a process does not exist density; one then has S(v) = in a rigorous mathematical sense. Indeed it can be interpreted as the derivative of the Wiener process, in a generalized function sense. (See Arnold [2, page for example.) This suggests that we consider the integrated version of our signal-noise equation:

J0000 ei27rvrV(T)dT).

V(T)

53],

Xt

t

1 s(u)du + Wt Wt

= = St +

8

Girsanov's Theorem

141

St

W s)ds,"

is a standard Wiener process and where is thought of as where the cumulative received signal. "J; x ( The key step in our analysis is the following consequence of the Girsanov­ Meyer Theorem.

on (0, F, IF, P), and Let be a standard Brownian motion W let H be bounded. Let t Xt = 1 Hsds + Wt and define Q by '!fj = exp{f(; -H8dW8 - � J(; H;ds} , for some T > 0. Then under Q, X is a standard Brownian motion for 0 < t < T. Proof Let Zr = exp{f:; - HsdWs - � Jt H;ds}. Then if Zt = E{Zri Ft} we know by Theorem 37 of Chap. II that Z satisfies the equation t Zt = 1 - lo Zs - HsdW8• Theorem 42. E lL

By the Girsanov-Meyer Theorem (Theorem 35) , we know that

t

Nt = Wt - 1 ;s d[Z, W]s

Q local martingale. However t [Z, W] t = [-Z_H W, W]t = 1 -Z8H8d[W, W] s t = - 1 Z8H8ds, since [W, W] t = t for Brownian motion. Therefore Nt = Wt - Jto - �Zs ZsHsds t = Wt + i H8ds = Xt, hence X is a Q local martingale. Since (J� H8ds)t>o is a continuous FV pro­ cess we have that [X, X]t = [W, W]t = t, and by Levy's Theorem (Theorem 39 of Chap. II) we conclude that X is a standard Brownian motion. Corollary. Let W be a standard Brownian motion and H E lL be bounded . Then the law of t Xt = 1 Hsds Wt, 0
·

0

+

oo ,

142

III Semimartingales and Decomposable Processes

Proof Let C[O, 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< t < T is a standard Brownian motion, it induces a measure J.Lw on C [O, T] :

P{w : t r+ Wt(w) E A}. Let J.Lx be the analogous measure induced by X. Then by Theorem 42 we J.L w ( A) =

have J.LX rv J.LW and further we have

0 Remark. We have not tried for maximum generality here. For example the

H

hypothesis that be bounded can be weakened. It is also desirable to weaken the restriction that E JL. Indeed we only needed that hypothesis to be able to form the integral J� This is one example to indicate why we need a space of integrands more general than JL.

H H8dW8•

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) = 0 a.e. , a.s.)? In terms of hypothesis testing, let denote the null hypothesis, the alternative. We have:

Ho

H1

Ho : Xr = Wr H1 : Xr = 1 H8ds + Wr. T

We then have

by the preceding corollary. This leads to a likelihood ratio test: if dw (w) < A, reject J.Lx

Ho,

if dw (w) > A , fail to reject J.Lx

H0,

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 be a standard Brownian motion on a space satisfying the usual hypotheses. Let fi (w, s, x) be functions satisfying (i = 1 , 2):

(D,.F,IF,P)

6

W

Stochastic "differential" equations have meaning only if they are interpreted stochastic integral equations.

as

9 The Bichteler-Dellacherie Theorem

(i) (ii) (iii)

143

y)i < Kl x - Yl for fixed (w, s) ; l fi(w,· s, x) -:F8fi(w,fors,fixed x);with right limits for fixed . fifi( w,s, x) is left continuous ( s, ( x) (w, x) ,

·,

E

By a Picard-type iteration procedure one can show there exists a unique so­ lution (with continuous paths) of

Xt = Xo + 1t h(·,s,Xs)dWs + 1t h (·,s,X8 )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 be any E L Define bounded, measurable function such that * * *

r(w, s, Xs) g (w, s,x) = h(w, s,x) + h(w, s,x)r(s,w,x).

,

r

We will see that we can find a solution of

provided we choose a new Brownian motion We define a new probability law Q by {T By Theorem

B appropriately.

dQ = ( r(s,Xs)dWs - 21 }{T r(s,X8) ds) . dP exp Jo 0 2

42 we have that

Bt = Wt - 1t r(s,Xs)ds

is a standard Brownian motion under Q. We then have that the solution ) also satisfies of

(

* * *

X

Xt = Xo + 1 t f1(-,s,Xs)dBs + 1 t (h + fn)(·,s,Xs)ds = Xo + 1t h ( s, Xs)dBs + 1t g(·, s, X8)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

Semimartingales 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 cadlag, the process Jt = L O<s l } has paths of finite variation on compacts; hence it is an FV process. Let Y = X - J. Then Y has bounded jumps. If we show that X is a classical semimartingale on [0, t0] , some t0 , then Y = X - J is one as well; moreover Y is special by Theorem 31. Let Y = Y0 M + A be the canonical decomposition of Y. Suppose t 1 > t0 and X is also shown to be a classical semimartingale on [0, tl] . Then let Y = Yo + N + B be the canonical decomposition of Y on [0, t 1 ] . We have that, for t < to,

+

A

M-N=B-A

by subtraction. Thus B is a predictable, FV process which is a local martingale; hence by Theorem 12 we have B = Thus if we take tn increasing to oo and show that X is a classical semimartingale on [0, tn] , each n, we have that X is a classical semimartingale on [0, oo ) . We now choose u0 > 0, and by the above it suffices to show X is a classical semimartingale on [0, u0] . Thus it is no loss to assume X is a total semimartin­ gale on [0, u0] . We will show that X is a quasimartingale, under an equivalent probability Q. Rao'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: -

where 0 = To given by

A.

n- 1 Ht = Hi 1 (r,, r,+,J i=O < T1 < · · · < Tn- 1 < Tn uo . In this case the mapping lx is

L

=

lx(H) = (H · X)u0 = Ho(Xr, - Xo) + · · · + Hn- 1 (Xuo - Xrn - I ) . The mapping Ix Su ----+ L 0 is continuous, where £0 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 (*) and I HI < '1 } .

Let f3 = lx(B), the image of B under lx. It will now suffice to find a probability Q equivalent to such that Xt E L 1 ( dQ ) , 0 < t < u0 and such that supU E f3 EQ (U) = c < oo . The reason this suffices is that if we take, for a given 0 = to < t1 < · · · < tn = u0 , the random variables

P

Ho = sign(EQ {Xt, - Xo i:Fo } ) , Hl that for this H E B,

9 The Bichteler-Dellacherie Theorem

= sign(EQ {Xt2 - X

it

I:Ft, } ), . . . ' we have

Since this partition T was arbitrary, we have Var(X) = supr Varr (X) supUE f3 EQ (U) = < oo, and so X is a Q quasimartingale.

c

Lemma 1. lim sup P( IYI C-+00

Y E {3

145

<

> c) = 0.

P I > c) > >

0. Then there ex­ Proof of Lemma 1. Suppose lim c->oo supY Ef3 ( I Y ists a sequence Cn tending to oo, Yn E /3, and a 0 such that P (IYn I > en) > a, all n. This is equivalent to p

c� l > 1) > a > 0.

Since Yn E /3, there exists Hn E B such that Ix(Hn ) = Yn . Then lx ( c: Hn ) = n tends to 0 uniformly a.s. which n ) _Ly _lJx(H n E /3, if Cn 1 . But _LH Cn Cn Cn implies that lx(.l... Hn ) = _Ly n tends to 0 in probability. This contradicts Cn Cn 0 (* ) .

>

=

Lemma 2. There exists a law Q equivalent to P such that Xt E L1 (dQ) ,

0 < t < uo .

Proof of Lemma 2. Let Y = supo
=

= P

=

Observe that f3 c L 1 ( dQ) for the law Q constructed in Lemma 2. Lemma 3 below implies that X is an R quasimartingale for R Q Hence by Rao's Theorem (Theorem 15) it is a classical R semimartingale, and by the Girsanov-Meyer Theorem (Theorem 35) it is a classical semimartingale for the equivalent law as well. Thus Lemma 3 below will complete the proof of Theorem 43. We follow Yan [233]. "'

"'

P.

P

Lemma 3. Let f3 be a convex subset of L 1 (dQ),

0 E /3, that is bounded in > 0 there exists a c > 0 such that Q(( > c) <

E, probability. That is, for any E for any ( E (3. Then there exists a probability R equivalent to Q, with a bounded density, such that supUE f3 ER (U) < oo.

Proof of Lemma 3 and end of proof of Theorem 43. To begin, note that the hypotheses imply that f3 c L 1 (dR). What we must show is that there ex­ ists a bounded random variable Z, such that P(Z 0) 1 , and such that sup( E f3 EQ (Z() < oo.

> =

III

146

Semimartingales and Decomposable Processes

A

Let E F such that Q(A) > 0. Then there exists a constant d such that Q(( > d) < Q(A)/2, for all ( E /3, by assumption. Using this constant let c = 2d, and we have that 0 < clA tt (3, and moreover if B+ denotes all bounded, positive r.v. , then clA is not in the (dQ) closure of /3-B+, denoted is L00 , and f3 - B+ is f3 - B+ . That is, clA tt f3 - B+ . Since the dual of 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

d,

L1

sup

(Ef3,TJEB+

EQ {Y(( - 17) }

L1

< cEQ {Y1A}.

Replacing 17 by a l {Y <0} and letting a tend to oo shows that Y > 0 a.s., since otherwise the expectation on the left side above would get arbitrarily large. Next suppose 17 = 0. Then the inequality above gives sup EQ {Y(} ( E (3

< cEQ{YlA } < +oo.

Now set H = {Y E B+ : sup(E (3 EQ {Y(} < oo } Since 0 E B+ , we know H is not empty. Let A = {all sets of the form {Z = O}, Z E H}. We wish to show that there exists a Z E H such that Q(Z = 0) = infAEA Q(A). Suppose, then, that Zn is a sequence of elements of H. Let Cn = sup( E f3 E{ Zn (} and dn = II Zn llv"' · (Since 0 E /3, we have Cn > 0). Choose bn such that 'L, bn Cn < oo and 'L, bn dn < oo, and set Z = 'L, bn Zn . Then clearly Z E H. Moreover, { Z 0 } = nn { Zn = 0}. Thus A is stable under countable intersections, and so there exists a Z such that Q( Z = 0) = infAEA Q(A). We now wish to show Z > 0 a.s. Suppose not. That is, suppose Q(Z = 0) > 0. Let Y satisfy ( ) (we have seen that there exists such a Y and that it hence is in H). Further we take for our set A in ( ) the set = { Z = 0}, for which we are assuming Q(A) > 0. Since 0 E f3 and 0 E B+ , we have from Lemma 2 that 0 < E{Y1A } = E{Y1 {z=o} } · Since each of Y and Z are in H, their sum is in H as well. But then the above implies .

=

*

*

A

Q{Y + Z = 0 }

= Q{Z = 0} - Q({Z = 0 } n {Y > 0}) < Q(Z = 0). This, then, is a contradiction, since Q(Z = 0) is minimal for Z E H. Therefore we conclude Z > 0 a.s., and since Z E B+ , it is bounded as well, and Lemma 3 is proved; thus also, Theorem 43 is proved. 0 We state again, for emphasis, that Theorems 26 and 43 together allow us to conclude that semimartingales (as we have defined them) and classical semimartingales are the same.

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. Rao [207] , Stricker [218] , and Metivier-Pellaumail [159] . 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 [40] . 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 [1 72] ; it was also proved independently by Doleans-Dade [51]. Le Jan's Theorem is from [112] . 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 [142] . The example from finance theory (starting on page 137) is inspired by [203] . Kazamaki's criterion was published originally in 1977 [1 24] ; see also [125] , whereas Novikov's condition dates to 1972 [184] . 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 [176] 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

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, lF, P) be a filtered probability space satisfying the usual hypotheses. Show that the predictable a-algebra (on JR.+ 0) is con­ Exercise 2. Let

x

tained in the optional a-algebra. ( Hint: Show that a cadlag, adapted process can be approximated by processes in lL.)

T be stopping times with S < T. Show that ( S, T] = S, t < T(w)} is a predictable set. Show further that [S, T) {andt (S, T) are optional sets. Last, show that if T is predictable, then (S, T) is a predictable set, and if both S and 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 Fr - = Vn Fsn . Exercise 7. A filtration lF is called quasi left continuous if for every pre­ dictable stopping time T one has Fr Fr- .7 Show that if lF 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 Fr = Vn Fsn . filtration such that there exists ( Note: One can have a quasi left continuous .)

Exercise 5. Let ( , w) : S(w) <

=

-+

a stopping time T with Fr -1- Fr _

Exercise 8. Show that the natural completed filtration of a Levy process is a quasi left continuous filtration.

} < and suppose M + A, with A predictable. Show

*Exercise 9. Let X be a semimartingale with E{[X, X] 00

oo

that X has a decomposition X = Xo + E{ [A , A]oo} < E{ [X, X] 00 } . ( Hint: Recall that if T is a predictable stopping time, then

6.Ar1{ T
predictable.)

t ) { {t )

t},

-1- 0}

( >l

Exercise 10. Let X and Y be semimartingales with the processes

[X, X] and

[Y, Y] locally integrable. Prove the following "sharp bracket" version of the Kunita-Watanabe inequality, for jointly measurable processes H, K:

( Warning : Recall that

when

(X, X)

(X, Y)

can exist as the compensator of and (Y, Y) do not exist.)

7 A quasi left continuous filtration is formally defined in Chap.

[X, Y]

even

IV on page 189.

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 (Tn ) n >I such that E{f0Tn JdA8 J} < oo.

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 Exercise 12. Let

functions of the interarrival times are absolutely continuous.

Exercise 13. Let T be an exponential random variable and

lF

=

l{t>T } ·

and suppose the filtration is constant: :Ft = :F0 for all t > 0. Show that the compensator A of N is At = l {t> T} · (This illustrates the importance of the filtration when calculating the compensator.) Let

:Fo

Nt

=

a{T}

N be a compound Poisson process, Nt = Li> l Ui 1{t> T,} 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{JUiJ} < oo and E{Ui } = 11 · Show that the compensator A of N for the natural filtration is given by At = AJ-lt. 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 (o:, {3), then its hazard rate is A(t) = o:{3cx t cx- 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 < AJ-l. 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 1 89 of Chap. IV.) Exercise 18. Let X be a semimartingale such that the process Dt = SUPs I of stopping times increasing to 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 1 32. Let Q rv P, and Zt = Ep { �� J:Ft } · Show that an adapted, dtdlag process M is a Q local martingale if and only if M is a P local martingale. Exercise 14. Let

00

Z

*Exercise 21. Let lF

C G be filtrations satisfying the usual hypotheses, and

let X be a G quasimartingale. Suppose that Yt = E{Xtl:Ft}, 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. Rao's Theorem .)

(Hint: Use

*Exercise 22. Suppose that A is a predictable finite variation process with E{[A, A]l } < oo , and that M is a bounded martingale. Show that [A, M] is 1

a uniformly integrable martingale.

Exercise 23. Let T be a strictly positive random variable, and let :Ft = a{T 1\ s; s < t}. Show that (Ft)t>O is the smallest filtration making T a stopping time.

Z be a cadlag supermartingale of Class D with Z0 = 0 and suppose for all predictable stopping times T one has E{�ZriFr- } = 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{6.Ar iFr - } = 0. 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{ Zrn } = E{Zr }. 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 Exercise 24. Let

regular.

*Exercise 27 (approximation of the compensator by Laplacians) . Let Z be a dtdlag positive supermartingale of Class D with limt-+ oo E{Zt} = 0. (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-+0 A�

=

Ar with convergence in £1 .

Z be a dtdlag positive supermartingale of Class D withh limt-+oo E{ Zt} = 0. Let Z = M- A be its Doob-Meyer decomposition. Let A be as given in Exercise 27. Show that for any stopping time T, limh-+O A� = Ar , but in this case the convergence is weak for £ 1 ; that is, the convergence *Exercise 28. Let

is in the topology a( £1 , £ 00 ). 8

Z be a cadlag positive supermartingale of Class D with limt-+oo E{ Zt} = 0. Let Z = M - A be its Doob-Meyer decomposition and assume further that A is continuous. Exercise 29 (discrete Laplacian approximations) . Let

Define 8

xn converges to X in cr (L 1 , U)() ) if xn, X are in L 1 and for any a.s. bounded random variable Y, E(XnY) -+ E(XY) .

Exercises for Chapter III

00

151

E{Z k A� = """' L-J

k=O

k k+ z , IF 2ft" � 2ft" }

Show that limn-+oo A� = A00 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 ]]N-' , and if X has continuous paths, then the filtration JFIL 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 (0, F,

lF, P) be the standard Brownian space. Show that

the optional a-algebra and the predictable a-algebra coincide. Meyer's Theorem (Theorem 4) and Exercise 30.)

(Hint:

Use

P) be a filtered probability space satisfying the usual hypotheses. Let X be a (not necessarily adapted) cadlag stochas­ *Exercise 32. Let (0, F, JF,

tic process such that for A > 0, E{f;' e - .At iXtldt} < oo. Let E{f000 e - >.s Xt+ sdsiFt}, the right continuous version. Show that

is an

lF martingale.

R.A(Xt)

*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 lL, the space of adapted processes with paths that are left continuous and have right limits. The space lL 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 H.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 Integrat ion 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 X0 = 0. This is a convenience involving no loss of generality. If Y is any semimartin­ gale we can set Yt = yt Y0, 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

1t HsdYs 1t Hsd� + Ho Yo . When Yo =f. 0, recall that we write J�+ HsdYs to denote integration on (0, t], and J� H8dY8 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.

p on JR.+

n is the smallest a­ E lL }. algebra making all processes in lL measurable. That is, P = a{ We let bP denote bounded processes that are P measurable.

Definition. The predictable a-algebra

X

H:H

X = M + A be a decomposition of a semimartingale X, with X0 = Mo = A0 = 0. Here M is a local martingale and A is an FV process (such Let

(

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

A

where N is a local martingale and 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 The 7t2 norm of X is defined to be X=N

+ A.

The space of semimartingales rf2 consists of all special semimartingales with finite 1{2 norm. In Chap. V we define an equivalent norm which we denote

Theorem 1 .

Proof.

I · IIH2 .

The space of 7t2 semimartingales is a Banach space.

The space is clearly a normed linear space and it is easy to check 2 that · 1i'H2 is a norm recall that E{N 00 } = E{[N, N] 00 }, and therefore II X IIH2 = 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 separately. Consider first N. Since E{N!, } = II [N, N] � 2 I I i2 , it suffices to show that the space of L2 martingales is complete. However an L 2 martingale M can be identified with Moo E L 2 , and thus the space is complete since L 2 is complete. Next suppose is a Cauchy sequence of predictable FV processes in p > 1 . To show converges it suffices II · where II A II v = II f000

I

(

A

l2

(An ) dA , l sii i LP

(An )

2 Stochastic Integration for Predictable Integrands

155

to show a subsequence converges. Therefore without loss of generality we can assume L n II An ll 2 < oo . Then L A n converges in · 11 1 to a limit A. Moreover

I

oo

r I dA� I = 0 L m-+ oo n >m Jo in L 1 and is dominated in L2 by Ln f000 IdA� I · Therefore L An converges to the limit A in II · ll 2 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 ) n >l is predictable, the limit A is the limit of predictably lim

D

measurable processes and hence also predictable.

For convenience we recall here the definition of JL.

Definition. lL (resp. blL) denotes the space of adapted processes with caglad 1 (resp. bounded, caglad) paths.

We first establish a useful technical lemma.

A be a predictable FV process, and let H be in lL such that E{f000 I Hs l l dAs l } < oo. Then the FV process (J� H8dA8)t>O is also pre­ Lemma. Let

dictable.

Proof.

We need only to write the integral J� H8dA8 as the limit of Riemann sums, each one of which is predictable, and which converge in ucp to J� H8 dA8 , showing that it too is predictable. D The results that follow will enable us to extend the class of stochastic integrands from blL to bP, with X E rt.2 (and Xo = 0). First we observe that if H E blL and X E H.2 , then the stochastic integral H · X E 7t 2 . 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]8(w) and J� IHs l l dAs l make sense for any H E bP and not just H E lL.

Definition. Let X E 7t 2 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 H2 the space blL 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 blL such that dx(H, J) < c}. Trivially A contains blL. If Hn E A and Hn increases to H with H bounded, then H E bP, and by the Dominated Convergence Theorem if J > 0 then for some N(J), n > N(J) implies dx(H, Hn ) < J. Since each Hn E A, we choose no > N ( J) and there exists J E blL such that dx ( J, Hno ) < J. Therefore given c > 0, by taking J = c/2 we can find J E blL such that dx(J, H) < c, and therefore H E A. An application of the Monotone Class Theorem yields the D

result.

Let X E H2 and Hn E blL such that Hn is Cauchy under dx . Then Hn X is Cauchy in 1{2. Proof. Since IIHn . x - Hm · X II H2 = dx(Hn , Hm ), the theorem is immediate. Theorem 3. ·

D

Let X E 1{2 and H E bP. Suppose Hn E blL and Jm E blL are two sequences such that limn dx ( Hn , H ) = limm dx(Jm , H) = 0. Then Hn · X and Jm · X tend to the same limit in 1{2. Proof Let Y = lim n-+oo Hn · X and Z = limm-+oo Jm · X, where the limits are taken in H2. For r:; > 0, by taking n and large enough we have II Y - Z IIH2 < II Y - H n · XIIH2 + II Hn · X - Jm · X ll w + IIJm X - ZIIH2 < 2c + IIHn · X - Jm · X II H2 < 2r:: + dx(Hn , Jm ) < 2r:: + dx(Hn , H) + dx(H, Jm ) < 4r::, Theorem 4.

m

0

and the result follows. We are now in a position to define the stochastic integral for H

X E H2) .

D

E bP (and

and let H E bP. Let Hn E blL Definition. Let X be a semimartingale in be such that limn -+ oo dx ( Hn , H) = 0. The stochastic integral H · X is the

(unique) semimartingale Y E write H o X = (J� H8dX8 ) t>O ·

H2

H2 such that limn-+oo Hn · X

=

Y in

H2. We

We have defined our stochastic integral for predictable integrands and semimartingales in 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.

1{2

2 Stochastic Integration for Predictable lntegrands

Theorem 5.

157

Let X be a semimartingale in H 2 • Then

Proof

For a process H, let H* = supt IHt l· 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}

l < 8E{[N , N] oo } + 2 ii fooo jdAs l ll i2 < 2E{ (N * ) 2 } + 2E{ ( oo ldAs l ) 2 } < 8 II X II�2 ·

0

Corollary. Let ( xn ) be a sequence of semimartingales converging to X in H2. Then there exists a subsequence ( ) such that limnk -> (Xnk - X)* = 0

nk

a.s.

oo

Proof

By Theorem 5 we know that (X n - X)* = supt IX;' - Xt l converges 0 to 0 in L 2 • Therefore there exists a subsequence converging a.s. 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 H 2 and H, K E bP. Then (H + K) · X = H · X + K · X,

and

H · (X + Y) = H · X + H

·

Y.

One need only check that it is possible to take a sequence Hn E blL that approximates H in both dx and dy. 0

Proof

Theorem 7. H · (XT ).

Let T be a stopping time. Then (H · Xf = Hl (o, TJ · X =

2 Indeed, it is an open question whether or not Theorem

integrands in

bP.

16 of Chap. II extends to

See the discussion at the end of this section.

158 Proof.

IV General Stochastic Integration and Local Times

Note that l [o , T] E b!L, so H l ro ,T] E bP. Also, X T is clearly still in H2 . Since we know this result is true for H E b!L (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)s)s > O is indistinguishable from

(Hs(LlXs))s> O · Proof. Recall that for a process J , Ll Jt = 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 b!L. Let H E bP, and let Hn E b!L such that limn --->oo dx (Hn , H) = 0. By the

corollary of Theorem 5, there exists a subsequence (

limoo ( Hnk · X - H · X)* nk -+ This implies that, considered as processes,

=

nk ) such that

0 a.s.

lim D.(Hnk · X) = D.(H · X), nk -+OO outside of an evanescent set.3 Since each Hn k E b!L, we have tl(Hnk · X ) = Hnk (tlX), outside of another evanescent set . Combining these, we have limoo Hnk (t::J. X )l { b. X �O } = lim b. (Hnk · X)l { b. X �O} nk -+OO nk -+ = D.(H . X)l { b.X�O } , and therefore

In particular, the above implies that limnk ---> oo Hf' k (w) exists for all (t, w) in { tlX =1- 0}, a.s. We next form

k (w) =f-Ht(w) and LlXt(w) =f-0}. A = {w : there exists t > O such that lim Hf' -+00 Suppose

nk

P(A) > 0. Then

00 { (LlNu ) 2 )}1!£2 dx (Hnk , H) > lil A { Jo (H'; k - Hs) 2 d( O< u <s 00 + I I 1 A { 0 I H'; k - Hs l d( L I!::J. Aul )}ll£2 , O< u <s and if LlXs =1- 0, then jtlNs l + jtlA l > 0. The left side of (*) tends to 0 and the right side of ( * ) does not. Therefore P(A) = 0, and we conclude tl(H · X) = limn k ---> oo Hn k t::J.X = HtlX. 0

L

1

as

nk

s

----> oo ,

3 A set

A C

JR+ x

the zero process.

n

is evanescent if

lA

is a process that is indistinguishable from

2 Stochastic Integration for Predictable lntegrands

159

X E rt2 , H E bP, and T a finite stopping time. Then H (xT- ) = (H · Proof By Theorem 8, (H · X f - = (H · X) T - Hr fl.Xrl{t> T} · On the other hand, x r - = X T - fl.Xrl{t> T } · Let At = fl.Xr l {t> T }· By the bilinearity (Theorem 6), H · (x T- ) = H · (XT ) - H · A. Since H · (XT ) = (H · X) T by Theorem 7, and H · A = Hr t:. Xr l { t> T } the result follows. D Corollary. Let

xf-.

·

•

The next three theorems all involve the same simple proofs. The result is known to be true for processes in blL; let (H n ) E blL approximate H E bP in dx ( · , ·), and by the corollary of Theorem 5 let n k be a subsequence such that nk lim (H nk -+00

· X H · X) * = O -

a.s.

Then use the uniform convergence to obtain the desired result. We state these theorems, therefore, without proofs.

H2

Theorem 9. Let X E 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) .

H2 and H · (K

Let X E H2 and H, K E bP . Then K · X E

X) = ( HK) · X. Theorem 11. Let X E be a (square integrable) martingale, and H E bP. Then H X is a square integrable martingale. Theorem 12. Let X, Y E and H, K E bP. Then ·

H2

·

H2

and in particular [H · X , H · X]t =

Proof

t

1 H;d [X, X] s,

(t > 0 ).

As in the proof of Theorem 29 of Chap. II, it suffices to show

t

[H · X, Y]t = lo Hsd[X, Y]s · Let (Hn ) E blL such that limn -+oo dx(Hn , H ) = 0. Let rm = inf{t > 0 : IY't l > m}. Then (rm) are stopping times increasing to oo a.s. and j YT I < m. 4 Since it suffices to show the result holds on [0, Tm ) , each m, we can assume without loss of generality that Y_ is in blL. Moreover, the Dominated Convergence Theorem gives limn dx (Hn y_ , HY_ ) = 0. By Theorem 29 of Chap. II, we have "'

..... oo

4

Recall that

y_

denotes the left continuous version of Y .

160

IV General Stochastic Integration and Local Times [Hn

· X, Y]t =

lo H-;d[X, Y]s t

(t > 0),

and again by dominated convergence

t

· X, Y] = { Hsd[X, Y]s (t > 0). lo It remains only to show limn _,oo [Hn · X, Y] = [H · X, Y] . Let zn Hn · X , be a subsequence such that limnk _, 00 ( zn k - Z)* = 0 a.s. , where and let Z = H · X (by the corollary to Theorem 5). Integration by parts yields [znk Y] = znk y (y_ ) znk ( znk ) . y = znk y - (Y_ Hn k ) X - (Znk ) . Y, limoo [Hn n -+

=

nk

,

_

.

_

.

where we have used associativity (Theorem 10). We take limits so that lim [znk , Y] nk -+OO

= ZY - y_ · (H · X) - z_ · Y = ZY - Y_ · (Z) - Z_ · Y = [Z, Y] = [H · X, Y].

0

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 7t2 . Fortunately Theorem 13 will show that 7t2 is merely a mathematical conve­ nience. Lemma. Let be an FV process with Ao = 0 and f000 ldAs l E L 2 . Then A E 7t 2 . Moreover IIAI!?-£2 < 6 11 f000 ldAs l ll £2 .

A

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} < oo. 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 sup ILt l

t

< k(Aoo + Aoo) + 2k L I�(A - A)sl s < 3k(A oo + Aoo) E L . -

1

Therefore L is a uniformly integrable martingale (Theorem 51 of Chap. I) and E{Loo} = E{Lo} = 0. Hence

2 Stochastic Integration for Predictable Integrands

161

E{Moo (A - A)oo} = E{[M, A - A]oo} = E{ [M, A]oo} - E{ [M, A]oo} = E{ [M, A] 00 }, because A is natural. By the K unit a-Watanabe inequality (the corollary to Theorem 25 of Chap. II)

E{l [M, A] ool} < (E{[M, M]oo} E{[A, A]oo} ) 1 12 < 21 E{[M, M]oo} + 21 E{[A, A]oo},

where the second inequality uses 2ab < a 2 + b2 . However

E{[M, M]oo} = E{M!} (Corollary 4 of Theorem 27 of Chap. II) and also [A, A] oo < A� a.s. Therefore 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� } < oo, A - A is a square integrable martingale, and IIA117-t 2 = II (A - A)ooll £2 + II Aoo ll £2 < 3IIAooll £2 for A increasing. 0 Remarks. The constant 6 can be improved to 1+VS < 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]). and

�

�

�

162

IV General Stochastic Integration and Local Times

In Chap. V we use an alternative norm for semimartingales which we denote II · II HP , 1 < p < oo . The preceding lemma shows that the norms II · 117-£2 and I I · II H2 are equivalent. The restrictions of integrands to bP and semimartingales to 1t 2 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 is said to hold locally for a process X if there exists a sequence of stopping times (Tn ) n >O such that 0 = T0 < T 1 < T2 < · · · < Tn < · · and limn oo Tn = oo a.s. , and such that xr n 1 { T >O } has property for each n. Since we are assuming our semi­ n martingales X satisfy Xo = 0, we could as well require only that xrn has property for each n. A related condition is that a property hold prelocally. 1r

_,

·

1f

1r

Definition. A property 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 oo a.s. such that xrn _ has property for each n > 1 . Recall that x r- = Xt 1 { oT } · The next theorem shows that the restriction of semimartingales to H2 is not really a restriction at all. 1r

1f

Theorem 13. Let X be a semimartingale, Xo = 0. Then X is prelocally in H2 . That is, there exists a non-decreasing sequence of stopping times (Tn ), limn -> oo Tn = oo a.s., such that xrn - E H 2 for each n.

Proof

Recall that xrn - = Xt 1 [o,T ) + Xrn _ 1 [T ,oo) · By the Bichteler­ Dellacherie Theorem (Theorem 43 of nChap. III) we ncan write X = M + A , where M is a local martingale and A is an FV process. By Theorem 25 of Chap. III we can further take M to have bounded jumps. Let f3 be the bound for the jumps of M. We define

Tn = inf{t > 0 : [M, M]t > n

t

lo JdAs l > n}

or

and let Y = xrn- Then Y has bounded jumps and hence it is a special semimartingale (Theorem 31 of Chap. III). Moreover .

or

Y = L + C,

where L = MTn and C = A Tn - - (�MT )1 [T ,oo ) · Then [L, L] < n + (32 , so n n 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 xrn -

=

L+C

E

H2 .

0

2 Stochastic Integration for Predictable lntegrands

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 b P . Definition. Let X E H2 with canonical decomposition X = N + A. We say H E P is (1t 2 , X) integrable if

Theorem 14. Let X be a semimartingale and let H E P be (H 2 , X) inte­ grable. Let Hn = H1 { 1HI
H2 . Proof.

Since H n E bP, each n, the stochastic integrals Hn · X are defined. Note also that limn -+oo Hn = H and that IHn l < IHI, each Then

I I Hn · X - Hm · X ll 1-t2 = dx (Hn , Hm ) 00 (H'; - H;") 2 d[N, N]s ) 1 12 ll £ 2 + II = 11 (

1

n.

00 1 I H'; - H;" lldAs l ll £2 ,

and the result follows by two applications of the Dominated Convergence D Theorem.

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 = H1 {1HI < 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 oo a.s. such that x rn _ E H2 , each > 1 , and such that H is (H 2 , X Tn _ ) integrable for each In this case we say H is X integrable, written H E L(X), and we define the stochastic integral by

n.

each

n.

Note that if m >

n then

n

IV General Stochastic Integration and Local Times

164

where Hk = H 1 {1HI< k }, by the corollary of Theorem 8. Hence taking limits we have H (X T"' - fn _ = H · (X Tn - ) , 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 1t 2 and such that H is (1t2 , X R£ - ) integrable, for each £. Again using the corollary of Theorem 8 combined with taking limits we see that ·

n

on [0, R£ 1\ Tn ), each £ > 1 and > 1 . Thus in this sense the definition of the stochastic integral does not depend on the particular sequence of stopping

times.

X,

If H E bP (i.e., H is bounded), then H E L(X) for all semimartingales 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 oo a.s. such that for each m > 1 , ( HtAs= 1 { S"'> 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.

Let X be a semimartingale and let H E P be locally bounded. Then H E L(X). That is, the stochastic integral H · X exists. Proof Let ( sm ) m > l , (Tn ) n > be two sequences of stopping times, each in­ and creasing to oo a.s., such that H8"' 1 { S"'>O } is bounded fo r each xrn _ E H2 for each n. Define Rn = min (sn , Tn ) nThen H = HRn 1 { Rn> O } on (O, Rn) and hence it is bounded there. Since X R - charges only (O, Rn), we have that H is (H 2 , X Rn - ) integrable for each n > 1 . Therefore using the Theorem 15.

1

m,

.

sequence Rn which increases to

oo

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"' - ) integrable, each and J is (H2 , xrn - ) integrable,n each n. Tak­ ing sn = Rn 1\ Tn , it is easy to check that o:H + (3J is (1t2 , X s - ) integrable m,

for each

0

n.

Let X, Y be semimartingales and suppose H E L(X) and H E L(Y). Then H E L(X + Y) and H · (X + Y) = H · X + H Y.

Theorem 17.

·

2 Stochastic Integration for Predictable lntegrands

165

Let X be a semimartingale and H E L(X). The jump process (b.(H X)s) s > O is indistinguishable from (Hs(b.Xs )) s > O · Theorem 19. Let T be a stopping time, X a semimartingale, and H E L(X). Then (H X f = Hl[o, T] · X = H · (X T ). Moreover, letting oo - equal oo , we have moreover

Theorem 18. ·

·

. Let X be a semimartingale with paths of finite variation on compacts. Let H E L(X) be such that the Stieltjes integral J� I Hs lldX8 I exists a.s., each t > 0. 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 = lot H.Ksd[X, Y]s (t > 0). Note that in Theorem 22 since H · X and H · Y are semimartingales, the

Theorem 20 .

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 > 0. 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 HQ · X denotes the stochastic integral computed under Q. By Theo­ rem 14 of Chap. II, we know that HQ · X = Hp · X for H E L, and therefore if X E H2 for both P and Q, they are equal for H E bP by the corollary of Theorem 5 . Let (Rf)£> 1 , (Tn ) n > l be two sequences of stopping times in­ creasing to oo a.s. such that H is (H2 , X Rt - ) integrable under Q, and H is n (H 2 , x r - ) integrable under P, each £ and n . Let sm = Rm 1\ Tm , so that H is (H2, x s = - ) integrable under both P and Q. Then H · X = limn _,oo Hn · X on [0 , sm ) in both dx(P) and dx(Q), where Hn = H l { I I n } E bP. Since D H? · X = H'Q · X, each n, the limits are also equal. H <

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

Proof

X E H 2 and X = N + A be its canonical decomposition. Then E{[N , N]oo} < E{[X , X] oo } ·

First observe that

[X, X] = [N, NJ + 2[N, A] + [A, A]. It suffices to show E{[N, A]oo} = 0, since then E{[N, N] oo } = E{[X, X]oo - [A, A]oo}, and the result follows since [A, A]oo > 0. Note that E{I [N, A]ool} < (E{[N, N]oo}) 112 (E{ [A, A]oo}) 1 12 < oo, by the Kunita-Watanabe inequalities. Also E{ [M, A]oo} = 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 _, 00 E{[Mn - N , M n - N] oo } = 0. Again using the Kunita-Watanabe inequalities we have

E{I [N - M n , A] ool} < (E{[N - Mn , N - Mn]oo}) 112 (E{[A, A]oo}) 112 and therefore limn -> oo E{[Mn , A]oo} = E{[N, A] 00 }. Since E{[Mn , A]oo } = 0, each n, it follows that E{[N, A]oo} = 0. 0 Note that in the preceding proof we established the useful equality E{[X, X] t} = 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 ,

II X I !?- t2 Proof

(H · X) :"x, ll £2 + 2II [X, X] � 2 II £2 < 5IIXI!?- t2· II HJ < l

< sup

I

By Theorem 5 for

!H I < 1

Since

oo

2 II [M , M] � 2 II £2 + 211 fo ldAs l ll £2 = 2IIXII 7-t 2 , where X = M + A is the canonical decomposition of X, we have the right inequality. For the left inequality we have II [M, M] � 2 II £2 < II [X, X] � 2 II £2 by the lemma preceding this theorem. Moreover if IHI < 1, then 2II [X, X] �2 II £2

<

2

Stochastic Integration for Predictable Integrands

167

II(H . A)t ll£ 2 < I I ( H . X)t ll p + II (H . M)t ll £2 < II( H . X) � ll£2 + II [M, M]�2 ll£2 ·

Next take H = 1�11 ; this exists as a predictable process since A is predictable, and therefore A predictable implies that I A I , the total variation process, is also predictable. Consequently we have

100

II X IIH2 = II [M, M]�2 II£2 + II ldAs l ll£ 2 = II [M, M] ZxS2 IIL2 + II ( H · A)oo l l£2 < II [M, M]�2 ll£2 + I I ( H . X) � ll£2 + II [M, M]�2 ll£2 < II (H . X) � ll £2 + 2 II [X, X]ZxS 2 II£2 .

0

We present as a corollary to Theorem 24, the equivalence of the two pseudonorms sup i H I < I II (H · X)�ll£ 2 and II X IIH2 · 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 < oo. (See also Dellacherie-Meyer [46, pages 303- 305] .) Corollary. For a semimartingale X (with Xo = 0),

II (H · X) � ll£2 < oo if and only if II X IIH2 < oo. Proof. By Theorem 5 if IIXI I H2 < oo and IHI < 1 we have

and in particular sup i H I < I

Thus we need to show only the left inequality. By Theorem 24 it will suffice to show that II [X, X] �2 IIL2 < sup II(H · X) � II£2 , I HI < l

for a semimartingale X with Xo = 0. To this end fix a t > 0 and let 0 = To < T1 < · · · < Tn = t be a random partition of [0, t] . Choose c: 1 1 , En non-random and each equal to 1 or - 1 and let H = 2::� Ei 1 (T,_, ,Ti ] . Then H is a simple predictable process and •

•

•

1

n (H · X)oo = L Ei (Xr, - Xr,_, ) . i= l Let o: = supi H I < I II (H · X ) � ll£ 2 · We then have 5

However the corollary does give some insight into the relationship between The­ orems 12 and 14 in Chap. V.

IV General Stochastic Integration and Local Times

168

n o: 2 > E{( H · X) �} = I: Ei E1 E{(Xr, - Xr,_1 )(Xrj - Xrj _ 1 ) } . i ,j=l

If we next average over all sequences c- 1 , . . { ±1 } n , we deduce

.

, En

taking values in the space

n n o:2 > I: E{ (Xr, - Xr,_1 )2 } = E{ I: (Xr, - Xr,_ 1 )2 } . i=l

i=l

Next let crm = {Tim } be a sequence of random partitions of [O, t] tending to the identity. Then Li (Xrt• - Xr;:'-. 1 )2 converges in probability to [X, X]t . Let { mk } be a subsequence so that L i (Xr,"'k - Xr;."_1 )2 converges to [X, X]t a.s. Finally by Fatou's Lemma we have

)2 } < o: 2 . "'k - Xr.1:nk E{""'(X E{[X, X] t} < limk ---7inf 1 r . 1 L...t 00 m i Letting t tend to

oo

we conclude that

E{[X, X]oo} < o:2 =

sup II (H · X) � lli2 . I HI < l

It remains to show that if sup i H I < I I I (H · X)� ll£2 < oo , then X E 1f2 . We will show the contrapositive. If X f/- 1i2 , then sup i H I < I II (H · X)� ll£2 = oo . Indeed, let Tn be stopping times increasing to oo such that x rn - is in 1i 2 for each (cf. , Theorem 13). Then

n

Letting

II XTn - IIH2 < 3

n tend to

oo

sup I I ( H · X Tn -) � 11£2 IHI < l

< 9II XTn - IIH2 ·

gives the result.

0

Before proving Theorem 25 we need two technical lemmas.

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, A00 = Ar- ) and let k be a constant such that Z < k on [0, T) . Then Lemma 1. Let

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 oo a.s. that reduce the local martingale (J� A8_dZs)t >o · Since dAs charges only [0, T) we have

t

t

1 Z8dA8 < 1 kdAs < kA00 for every

t > 0. Therefore E{(AZ) Rn } = E{

1

Rn

A8_dZs} + E{

1

Rn

Z8dAs}

< 0 + E{kAoo } · The result follows by Fatou's Lemma.

0

X be a semimartingale with Xo = 0, let Q be another prob­ ability with Q � P, and let Zt = Ep {� I .Ft}· If T is a stopping time such that Zt < k on [0, T) for a constant k, then Lemma 2. L et

Proof By Theorem 24 we have II Xr - ll ?t2(Ql < sup EQ {((H · x r -) * )2} 1 + 2EQ{[xr - , x r - ] 4 } IHI < l

<

� ���1 Ep { �� ((H · xr- )*)2 }

< Vk sup

IHJ < l

1 2

+ 2Ep

{ �� [xr- , xr-l � }

Ep{((H x r -) * )2} � + 2 VkEp { [XT- , xr - ] � }. ·

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 1i2(P) implies that X E 1i2 (Q) as well, with the estimate II X II?t2 ( Q) < 5v'kiiX II?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

IHI
EQ {((H . X) �Y } = <

where again

k is the bound for ��.

sup

I HI < l

k

dQ ((H . X) ) 2 } Ep { dP �

sup

IHI < l

Ep{((H X )� ) 2}, ·

170

IV General Stochastic Integration and Local Times

Let X be a semimartingale and H E L(X). If Q � P, then H E L(X) under Q as well, and Hq · X = Hp · X, Q-a.s. Proof Let rn be a sequence of stopping times increasing to oo a.s. such that H is ( XTn _ , 1{.2) integrable under P, each n > 1. Let Zt = Ep{�IFt }, the cadlag version. Define sn = inf { t > 0 : I Zt I > n} and set Rn = sn 1\ rn. Then X Rn - E 1i2(P) n 1i2 ( Q ) by Lemma 2, and H is (1i2, X Rn -) integrable under P. We need to show H is (1i2 , X Rn - ) 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 = H1 { IHI<m } · Then Theorem 25.

/

J I H';'I / dCsi i P (Q)

(Eq{ (Hr;' ) 2 d [N, N]s }) 112 + 1/ = IIHm . x R " - II?t2(Q) < 5 vfnii Hm . x Rn - ll?t 2( P) < 5y niiH · X Rn - �� ?t2 (P) < oo, and then by monotone convergence we see that H is (1i2, X Rn - ) integrable under Q. Thus H E L(X) under Q, and it follows that Hq · X = Hp . X, r::;

0

Q-a.s.

Theorem 25 can be used to extend Theorem 20 in a way analogous to the extension of Theorem 1 7 by Theorem 18 in Chap. II.

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) > 0. 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 H and H as well as X and X are indistinguishable. Thus Hq ·X = Hq X and hence E-X = H · X P-a.s. on A by Theorem 25, since Q � P. Theorem 26.

.

·

The second assertion has an analogous proof (see the proof of Theorem 1 8 of Chap. II) . o

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 1 8 of Chap. II.

Corollary. With the notation of Theorem 26, let S, T be two stopping times

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)}. T Then H x r - H · X8 = H · X - H · X8 on C and H · xr - H · X8 equals a path-by-path Lebesgue-Stieltjes integral on D. with

·

2 Stochastic Integration for Predictable Integrands

171

Let Pk be a sequence of probabilities such that X is a Pk semimartingale for each k. Let R = I:;;' 1 >.. k Pk where A k > 0, each k, and I:%" Ak = 1 . Let H E L(X) under R. Then H E L(X) under Pk and HR · X = Hp• · X, Pk -a.s., for all k such that Ak > 0. Proof. If Ak > 0 then Pk « R. Moreover since Pk (A) < }. R(A), it follows that H E L(X) under Pk . The result then follows by Theorem 25. 0 Theorem 27. 1

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). ·

M be a square integrable martingale and let H E P be such 00 that E{f0 H'1d[M, M]s } < oo. Then H · M is a square integrable martingale. Proof. If H k E bP , then H k M is a square integrable martingale by The­ orem 11. Taking H k = H1{1 H I
·

grable martingale.

0

Theorem 28. Let M H E P. The stochastic

be a locally square integrable local martingale, and let 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 > 1 increasing to oo a.s. such that E{f0Tn H';d[M, M]s } < oo.

Proof. We assume that M is a square integrable martingale stopped at the time Tn . The result follows by applying the lemma. o Theorem 29. Let M be a local martingale, and let H E 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 • k R there exists R increasing to oo a.s. such that M - E 1i 2 , and since H E b P • • there exist processes Hl E blL such that IIH£ MR - - H M R - II1t 2 tends • • to zero. In particular, Hl MR - te nds to H MR - in ucp. Therefore we • can take Hl such that Hl · MR - tends to H M in ucp, with Hl E blL . Finally without loss of generality we assume Hl · M converges to H · M i n ucp. Since Hl · M = Hl N + Hl A and Hl · N converges to H · N in ucp, we deduce Hl A converges to H · A in ucp as well. Let 0 < s < t and assume Y E b.F8• Therefore, since A is of integrable total variation, and since Hl · A is a martingale for Hl E blL ( see Theorem 29 of Chap. III), we have ·

·

·

·

·

·

·

·

172

IV General Stochastic Integration and Local Times

1

t t lim H�dAu} = E{Y lim E{Y HudAu} = E{Y l->oo t->oo s+ s+ = lim E{Y

1

l->oo

= 0,

t H�dAu} s+ t H�dAu} s+

1 j

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 (f� H8dAs)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. D ·

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 H E 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

MT - is a local martingale. Proof. The notation MT- means

stopping time. Then

Mr- = Mt 1 {t < T } + Mr _ 1 {t > T} > where M0 _ = M0 = 0. Let (Sn ) n > l be a sequence of stopping times announc­ ing T. On {T > 0}, lim =1 n---+cx:> 1 [0' sn ] [0 T) , and since 1 [o , s ] is a left continuous, adapted process it is predictable; hence n l [o , T) is predictable. But f� 1[o ,r) (s)dMs = Mt - by Theorem 18, and hence it is a local martingale by Theorem 29. 0 '

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 MT- is still a local martingale. Since a continuous local martingale is locally square integrable, the theory is particularly nice.

Let M be a continuous local martingale and let H E P be such that I; H;d[M, M]s < oo a.s., each t > 0. Then the stochastic integral H · M exists (i.e., H E L(M)) and it is a continuous local martingale. Proof Since M is continuous, we can take Rk = inf{t > 0 : I Mt l > k}. Then I Mti\ Rk I < k and therefore M is locally bounded, hence locally square integrable. Also M continuous implies [M, M] is continuous, whence if t yk = inf{t > 0 : lo H_;d[M, M ] s > k},

Theorem 30.

we see that u; H_;d[M, M]s)t >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 !l(H·M) = H(!lM) and 11M = 0 by hypothesis. D

In the classical case where the continuous local martingale M equals B, a standard Brownian motion, Theorem 30 yields that if H E P and I H; ds < 00 a.s., each t > 0, then the stochastic integral (H . Bt)t >O = HsdBs)t>O exists, since [B, B]t = t.

u�

�

X be a continuous semimartingale with ( unique) decomposi­ tion X = M + A. Let H E P be such that t t H'1d[M, M]s + I Hs l ldAs l < 00 a.s. Corollary. Let

lo

lo

each t > 0. Then the stochastic integral continuous.

(H · X)t =

I� H8dXs exists and it is

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 D continuous, and the result follows from Theorem 20. In the preceding corollary the semimartingale X is continuous, hence [X, X] = [M, M] and the hypothesis can be written equivalently as

t

t

lo H'1d[X, X]s + lo I Hsl l dAs l < 00

a.s.

each t > 0. 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

Let M be a local martingale with jumps bounded by a constant (3. Let H E P be such that I� H,?d[M, M]s < oo a.s., t > 0, and E{H:j.} < oo for any bounded stopping time T. Then the stochastic integral (J� H8dMs) t >O exists and it is a local martingale. Proof Let Rn = inf{t > 0 : I� H,?d[M, M]s > n }, and let rn = min(Rn , n). Theorem 31.

Then rn are bounded stopping times increasing to oo a.s. Note that

E{

fo

Tn

H,?d[M, M] s} < n + E{ Hfn (LlMr n )2} < n + !P E{Hf } < oo, n

and the result follows from Theorem 28.

0

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 Hm E P be a sequence converging a.s. to a limit H. If there exists a process G E L(X) such that I H m l < G, all m, then Hm , H are

in L(X) and Hm · X converges to H · X in ucp. Proof. First note that if I J I < G with J E P, then J E L(X). Indeed, let (Tn ) n > l increase to oo a.s. such that G is (1i2, X Tn -) integrable for each n.

Then clearly

E{

1

00

1

00

J;d[N, N]s} + E{( I Js l ldAs l )2 } 00 00 < E{ a;d[N, N]s} + E{ ( I Gs l ldAs i )2} < oo ,

fo

fo

and thus J is (1i 2, x rn - ) integrable for each n. (Here N + A is the canonical decomposition of xrn - .) 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 > 0 be given, and choose n such that P ( Tn < to) < c, where x rn - E 1i2 and G is (1i 2, x rn - ) integrable. Let x rn - = N + A, the canonical decomposition. Then E {sup ! H m . x rn

t < to

_ - H . x rn - 1 2}

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

E { sup I (Hm - H) · Nj2 }

t < to

175

< 4E{ j (Hm - H) · Nto l 2 }

and again this tends to zero by the Dominated Convergence Theorem. Since convergence in L2 implies convergence in probability, we conclude for c5 > 0, limsupP{ sup I Hm · Xt - H · Xt l > c5} t < to m -+<XJ < lim sup P{sup i ( (Hm - H) · xr n - )ti t
E

is arbitrary, the limit is zero.

> c5} + P (Tn < to) 0

We use the Dominated Convergence Theorem to prove a seemingly in­ nocuous result . Generalizations, however, are delicate as we indicate following the proof.

Let lF = (Ft)t>O and G = Wt ) t >o be two filtrations satisfying the usual hypotheses and suppose Ft C 9t , each t > 0, and that X is a semimartingale for both IF and G. Let H be locally bounded and predictable for lF. Then the stochastic integrals Hw · X and HG · X both exist, and they are equal.6 Proof. It is trivial that H is locally bounded and predictable for (Qt)t >o as Theorem 33.

well. By stopping, we can assume without loss of generality that H is bounded. Let

1i = {all bounded, :F predictable H such that Hw · X = HG · X } . Then 1i is clearly a monotone vector space, and 1i contains the multiplicative class blL 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 1 6 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 9t , 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 Wt)t� o, respectively.

IV General Stochastic Integration and Local Times

176

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 = (Xt) t >o be a stochastic process given by the following description. Let T be an exponential random variable with parameter ,\ = 1 , let U be an independent random variable such that P { U = 1 } = P { U = 1 } = 1/2, and set X = U1 {t >T } . Then X together with its minimal filtration satisfying the usual hypotheses is a martingale in 1i2. That is, X is a stopped compound Poisson process with mean zero and is an L2 martingale. Let Ht = ! 1 {t>O} · Therefore H is a deterministic integrand, continuous on ( O, oo ) , and hence predictable. Consequently the path-by-path Lebesgue-Stieltj es integral Zt = J� H8 dX8 exists a.s. However H · X is not locally in JiP for any p > 1 . ( However since it is still a semimartingale7 , it is prelocally in 1i2.) Moreover , even though X is an L2 martingale and H is a predictable integrand, the stochastic integral H · X is not a local martingale because E{ I Zs l } = oo for every stopping time S such that P(S > 0) > 0. -

The next theorem is useful, since it allows one to work in the convenient space 1i2 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 JiP for any p > l.

Let X be a semimartingale on a filtered complete probability space (n, :F, lF, 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 1i2. Moreover, �� can be taken to be bounded.

Theorem 34.

Before we begin the proof we establish a useful lemma.

(D, :F, P)

be a complete probability space and let Xn 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 Xn is in L2( dQ ) .

Lemma. Let

Proof.

Assume without loss that each of the Xn is positive. For a single ran­ dom variable X we take Ak = { k < X < k + 1 } and Y = L k > I T k 1 Ak · Then Y is bounded and

EQ{X2} = E{X2Y} < 2:.::: ( k ;k1)2 < oo . k> l For the general case, choose constants O:n such that P(Xn > o:n ) < 2- n . The Borel-Cantelli Lemma implies that a.s. Xn < O:n for all n sufficiently large. Next choose constants Cn such that Ln > l Cn O:n < oo. 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 Predic table Integrands

177

density chosen for Xn individually as done in the first part of the proof, and take Y = Ln >l cnYn so that we have the result. D 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
n.

lo

where Zt = Ep{ZIFt} with Z = �� · For convenience we write this as Ct = At + vt . We now need to establish that Eq {(f; l dVs l) 2 } = Eq{IV I D < oo. Note that we have Eq {V?} < oo , since vt = Mt - At, and At E L 2 (dQ). It is easy to check that Ep{ [M, M]n} < 4Ep { [X, X]n} < oo by an argument similar to the proof of the lemma preceding Theorem 24. Finally L2 (dP) c L 2 (dQ). Since V is predictable, there exists a predictable process H taking values in { -1, 1} such that J; I dV8 I = J; H8dV8• Setting S = H X, we have [S, S] = [X, X], and the random variables [S, S]n are then integrable (dP). S has a decomposition S = L + D where L = H M and D = H · A, hence under the law Q we have L = H · N + H · V, always using that the stochastic integral is invariant under a change to an equivalent probability measure. We deduce that EQ {(f; H8dV8) 2 } < oo. Since Eq {(f; H8dV8) 2 } = Eq{ (f� ldV81) 2 }, coupled with I H Alt = I Ait, the result follows. D Corollary (Lenglart's Inclusion Theorem) . Let (n, .F, IF, P) satisfy the usual hypotheses, and suppose Q is an equivalent probability measure with a bounded density with respect to P. Then 1t2 (P) c 1t2 (Q). That is, 1t2 with respect to P is contained in the space of 1t2 semimartingales with respect to ·

·

·

Q. Proof. Let X be an 1t2 semimartingale with respect to P. As we saw in the proof of Theorem 34, since Q has a bounded density, we obtain that [X, X] is in L 1 (dQ), and hence X is special for Q. We then obtain a canonical decomposition of X and show both terms are in 1t2 analogously to how we did it in the proof of Theorem 34.

D

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 (f!, :F, (:Ft)t>o, 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. of L 2 martingales is all martingales M such that supt E { M[} < oo, and Mo = 0 a.s. Notice that if M E M2 , then lim t---. oo E{Ml } = E{M!} < oo , and Mt = E{Moo i :Ft } . Thus each M E M2 can be identified with its terminal value M00 • We can endow M2 with a norm

Definition. The space

M2

IIM II = E{ M�j 1/ 2 = E{[M, M] 00 } 1 1 2 ,

and also with an inner product

(M, N) = E{MooNoo}, for M, N E M2 . It is evident that M2 is a Hilbert space and that its dual space is also M2 . If E{M[} < oo for each t, we call M a square integrable martingale. If in addition supt E{M[ } = E{M!} < oo, then we call M an 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 = 0. If we did not have this hypothesis, we would have to add the condition that for any event A E :F0 , any martingale M in the subspace F, then Ml11 E F.

Definition. A closed subspace F of M2 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 MT E F).8 Theorem 35. Let F be a closed subspace of M2 . Then the following are

equivalent. (a) F is closed under the following operation. For M E F, (M - Mt )lA E F for A E :Ft, any t > 0. (b) F is a stable subspace. 8

Recall that M[ = Mti\T, and Mo = 0.

1 79 (c) If M E F and H is bounded, predictable, then (f� H8dMs)t> o = H · M E F. (d) If M E F and H is predictable with E{f000 H;d[M, M]s} < oo, then H · M E F. Proof. Property (d) implies (c) , and it is simple that (c) implies (b) . To get 3 Martingale Representation

(b) implies (a) , let T = t11 , where t11

=

{

t, oo ,

if w E A, if w fj_ A.

Then T = t 11 is a stopping time when A E :Ft, and (M - Mt )lA = M - MT ; 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

Ht = 1Aa 1 {o} + L 1 Ai 1(t, ,t,+tl i=l

with Ai E :Ft, , 0 < i < n, 0 = to < t 1 < · · · < tn+ I < oo, then H · M E F whenever M E F. Linear combinations of such processes are dense in blL which in turn is dense in bP under dM( ·, ·) by Theorem 2. But then bP is dense in the space of predictable processes such that E{f000 H;d[M, M]s} < oo, as is easily seen (cf., Theorem 14) . Therefore (a) implies (d) and the theorem is proved. D

1{

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 M2 with its terminal value Moo E L2 . Therefore another martingale N E M 2 is (weakly) orthogonal to M if E{N00 M00 } = 0. There is however another, stronger notion of orthogonality for martingales in M2 .

Definition. Two martingales N, M E M2 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 NM 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 M2 are strongly orthogonal if and only if [M, N] is a uni­ formly integrable martingale. If N and M are strongly orthogonal then E{N00 M00 } = E{Loo} = E{Lo} = 0, so strong orthogonality implies or­ thogonality. The converse is not true however. For example let M E M 2 , and

180

IV General Stochastic Integration and Local Times

let Y E :Fo, independent of M, with Nt = YMt , t > 0. Then N E M2 and

P(Y

=

1)

=

P(Y

= -1)

�- Let

E{Noo Moo} = E{YM�J = E{Y}E{M!} = 0, so M and N are orthogonal. However MN = YM2 is not a martingale (unless M = 0) because E{YM[ I:Fo} = YE{M[ I :Fo} =f- o = YM;?.

A of M2 we let A_!_

(resp. Ax ) denote the set of all elements of M 2 orthogonal (resp. strongly orthogonal) to each element of A.

Definition. For a subset

Lemma 1. If A is any subset of M2 , then Ax is (closed and) stable.

Proof Let Mn be a sequence of elements of Ax converging to M, and let N E A. Then Mn N is a martingale for each n and Ax will be shown to be closed if MN is also one, or equivalently that [M, N] is a martingale. However E{I [Mn , N] - [M, N] t l } = E{I [Mn - M, N] tl } < (E{[Mn - M, Mn - M] t } ) 11 2 (E{[N, N] t } ) 11 2 by the Kunita-Watanabe inequalities. It follows that [Mn , N] t converges to [M, N]t in L 1 , and therefore [M, N] is a martingale, and Ax is closed. Also Ax is stable because M E Ax , N E A implies [MT , N] = [M, Nj T is a martingale and thus MT 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 M E Ax . 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 MT which is in S(M ). 0

Let M 1 , . . . , Mn E M2 , and suppose Mi , MJ are strongly or­ thogonal for i =f- j. Then S(M1, . . . , Mn ) consists of the set of stochastic integrals n H 1 . M 1 + . . . + Hn . Mn = 2: Hi . Mi ' i= 1

Theorem 36.

3 Martingale Representation

181

where Hi is predictable and 1
.

n

-

n.

Let I denote the space of processes L i=l H' · M' , where H' 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

Proof.

.

.

Then the mapping (H 1 , H2, . . . , Hn ) --> L� 1 Hi Mi is an isometry from Llr1 EB · · · EB Llr n into M2. Since it is a Hilbert space isometry its image I is complete, and therefore closed. D ·

Let A be a subset of M2 which is stable. Then AJ.. is a stable subspace, and if M E AJ.. then M is strongly orthogonal to A. That is, AJ.. = Ax , and S(A ) = AJ..J.. = Ax J.. = Ax x . Proof We first show that AJ.. = Ax . Let M E A and N E AJ... Since N is or­ Theorem 37.

thogonal to S(M) , by Lemma 2, N and 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 M2, 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 D A, and we conclude A is a stable subspace.

A be a stable subspace of M2 . Then each M E M2 has a unique decom position M = A + B, with A E A and B E Ax . Proof A is a closed subspace of M2 , 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 Corollary 1. Let

Theorem

37.

D

Corollary 2. Let M, N E

M2 , 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. ·

We know that such an L exists by Corollary 1 . Since { M} consists of 0 just one element we can apply Theorem 36 to obtain the result.

Proof

182

IV General Stochastic Integration and Local Times

Definition. Let A be finite set of martingales in M2 . We say that A has the (predictable) representation property if I = M2 , where I

= {X : X = L Hi . M i ' Mi E A} ' n

i =l

each Hi predictable such that

Corollary 3. Let A = { M l , . . . , Mn }

M 2 , and suppose Mi , M1 are strongly orthogonal for i =f- j. Suppose further that if N E M2 , N _l A in the strong sense implies that N = 0. Then A has the predictable representation property. C

Proof. By Theorem 36 we have S(A) = I. The hypotheses imply that D S(A)J. = {0}, hence S(A) = M2 •

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 (n, :F, (:Ft ) t >O, P) , and a set of martingales A in M2 . We will see that the property that S(A) = M2, 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 L2 (:Ft h>o martingales of all the elements of A. Our first observation is that since the filtration IF = (:Ft ) t >O is assumed to be P complete, it is reasonable to consider only probability measures that are absolutely continuous with respect to P. 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< t< oo :Ft such that

Definition. Let A

(i) Q « P, (ii) Q = P on :Fo , (iii) if X E A then X E M2 (Q),

where M2 (Q) denotes all (Q, IF) L2 martingales.9

Lemma. The set M2 is a convex set. 9

N ote that while lF satisfies the usual hypotheses under P, :F0 does not contain all

of the Q-null sets, and thus lF does not completely satisfy the usual hypotheses under Q. This does not create any problems, however.

3 Martingale Representation

183

Proof Let Q and R E M 2 (A) , and let S = AQ + (1 - A)R, 0 < A < 1 . Then

for X E M 2 (A) , < (1 A)ER{M } = sup Eq {M oo , sup Es{M } } [A ( ( + ] ( t t since Q, R E M 2 (A) . Also if H E b.F8, s < t, then Es{MtH} = AEq {MtH} + ( 1 - A)ER{MtH} = AEq {MsH} + (1 - A)ER{MsH} = Es {MsH} ,

D

Definition. A measure Q E M2 (A) is an extremal point of M 2 (A) if whenever Q = AR + (1 - A)S with R, S E M 2 (A), R -=/:- S, 0 < A < 1 , then A = 0 or 1 . Theorem 38. Let A C M2 . If S(A) = M2 then P is an extremal point of M 2 (A) .

Proof. Suppose P is not extremal. We will show that S(A) -=/:- M2 . Since P is not extremal, there exist Q, R E M 2 (A) , Q -=/:- R, such that P = AQ + (l - A)R, 0 < A < 1 . Let dQ Loo = dP ' and let Lt = E{ � I.Ft}. Then 1 = �� = AL00 + (1 - A) �� > AL00 a.s., so that L00 < i a.s. Therefore L is a bounded martingale with L0 = 1 (since Q = P on .F0), and thus L - L 0 is a nonconstant martingale in M2 (P). However, if X E A and H E b.F8, then X is a Q martingale and for s < t, Ep{XtLtH} = Ep{XtLooH} = Eq {XtH} = Eq {XsH} = Ep{ XsLoo H} = Ep{XsLsH}, and XL is a P martingale. Therefore X( L - L0) is a P martingale, and L - L 0 E M2 and it is strongly orthogonal to A. By Theorem 37 we cannot D have S(A) = M2 .

Let A C M2 . 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 Loo Loo 1 ( )dP and dR = (1 dQ = + 2c )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 Theorem 39.

184

IV General Stochastic Integration and Local Times

Let A = { M 1 , . . . , Mn } C M 2 , with Mi continuous and Mi, MJ strongly orthogonal for i -1- 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 < oo ) > 0. By Theorem 22 of Chap. III, there exists a martingale M with l:!.MT = l {T 0 : I Mt l > n } , I M Rn I < n + 1 and thus M is locally bounded. Indeed, we then have MRn are bounded martingales strongly orthogonal to Mi . By Theorem 39 we have MRn = 0 for each n. Since limn _.oo Rn = oo a.s., we conclude M = 0, a Theorem 40.

contradiction. (b) Let M be a bounded martingale which is not continuous, and assume Mo = 0. Let ye: = inf{t > 0 : l llMt l > c}. Then there exists c > 0 such that for T = re: , P{ I L:!.MT I > 0 } > 0. By part (a) the stopping time T is accessible, hence without loss we may assume that T is predictable. Therefore MT - is a bounded martingale by the corollary to Theorem 29, whence N = MT - MT - = LlMT l {t > T } 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 = MT - MT- = 0 by Theorem 39, and we conclude that M is continuous. (c) Let M be a uniformly integrable martingale closed by M00• Define

Then Mn are bounded martingales and therefore continuous by part (b). However 1 < E {I M� - Moo ! } M c} P{sup I M;' - t l >

t

c

by an inequality of Doob 1 0 , and the right side tends to 0 as n tends to oo . Therefore there exists a subsequence (n k ) such that lim k _.oo M;'k = Mt a.s., uniformly in t. Thus M is continuous. (d) By Corollary 3 of Theorem 37 it suffices to show that if N E Ax then N = 0. Suppose N E Ax . Then N is continuous by (c) . Therefore N is locally D bounded, and hence by stopping, N must be 0 by Theorem 39. 10

See, for example, Breiman [23, page 88] .

185

3 Martingale Representation

The next theorem allows us to consider subspaces generated by countably infinite collections of martingales.

Let M E M2, yn E M2, n > 1, and suppose Y� converges to Y00 in L2, and that there exists a sequence Hn E L(M ) such that �n = I� H�dM8, n > 1. Then there exists a predictable process H E L(M) such that yt = I; H8dM8 • Proof If Y� converges to Y00 in L2, then yn converges to Y in M2. 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. D Theorem 42. Let A = { M 1 , M2 , . . . , Mn , . . . } , with Mi E M2, and suppose there exist disjoint predictable sets Ai such that 1A , d[Mi , Mi ] = d[Mi , Mi ], i > 1. Let At = L� J; 1A, (s)d[M i , Mi ]s · Suppose that (a) E{Aoo } < oo; and (b) for :P c :Foo such that for any Xi E b:Fi , we have Xf = E{X i \:Ft } = I� H! dM;, t > 0, for some predictable process Hi . Then M = L� Mi exists and is in M2, and for any Y E b V i P , if yt = E{Y\:Ft}, we have that yt = J� H8dM8, for the martingale M = L� M i and for some H E L(M). Proof. Let Nn = L� Mi . Then [Nn , Nn ] t = L� 1 f; 1A, (s)d[Mi , M i ]s, hence E{(N:;_, )2} = E{[Nn , Nn ]oo} < E{A00}, and Nn is Cauchy in M2 with limit equal to M. By hypothesis we have that if Xi E b:Fi then t t E{X i \:Ft} = Xf = lo H;dM; = lo 1A' H;dM; t = { 1 A' H; d(M; + 2: MJ ) i #J t = lo 1A,H!dM8• Theorem 41.

1

1

1

1

lo

Therefore if i

-=/=- j we have that

t

lo 1A'H! 1Aj HJd [M, M]s t = 1 1 A'nAj H;H1 d [M, M]s

[Xi , X1 ] =

= 0,

since Ai n AJ

= 0, by hypothesis. However using integration by parts we have

186

IV General Stochastic Integration and Local Times

t t x;x; = lo x;_ dxt + fo xt_dx; + [Xi , xJ ] t t t J dx i i dXJs + 1 xs= j xss 0 0 t t = lo x;_ lJ1j HtdMs + lo XL lAiH�dMs t = 1 Hi,J dM 0

S

Sl

where Hi ,j is defined in the obvious way. By iteration we have predictable representation for all finite products ft
X = (X1, and let IF = (.Ft)O
locally square integrable local martingale M for IF has a representation where Hi is (predictable, and) in L (X i) . Proof. Fix t0, 0 < t < to , and assume

= {X 1, [Xi, X1]

X2 is stopped2 atbeto.allThen letting

probability A . . . , xn }, we have that A c M • Let M (A) measures Q such that Q � P, and under Q all of A C M 2 (Q) . U Since is the same process under Q as under P we have by Levy's Theorem ( 1 , . . . , xn ) is a Q-Brownian motion (Theorem 40 of Chap. II) that as well. Therefore, for bounded Borel functions /I , . . . , fn , we have that for f1� / have the same expectation t 1 < t 2 < tn < to, /I and we conclude that M 2 (A) for Q as they do for P. Thus P = Q on is the singleton P}. The probability law P is then trivially extremal for M 2 (A) , and therefore by Theorem 40 we know that A has the predictable representation property. k Suppose M is locally square integrable and = M l\to . Let (T ) k >O be 0, and such that a sequence of stopping times increasing to oo a.s., T0 MT k E M 2 , each k > 0. Then by the preceding there exist processes Hi , k such that n MTk 2: i k . x i , each k > o . ·

·

·

{

X= X (XtJ /i(Xt;+1 -Xt;) .Ft0,

Mt

= H i=l

11

,

Recall that M2(Q) denotes all L2 Q martingales.

t =

3 Martingale Representation

187

n M = � Hi · X i . i=l

Next let (te)e> o be fixed times increasing to oo with to = 0. For each te we know there exist Hi ,£ such that for a given locally square integrable local martingale N

n Nt l'lt t = '""" � (Hil'' · X i )t l'ltt · i =l By defining H i = '5:.':: 1 H i, l' l ( tt _ 1 , tt ] one easily checks that Hi E L(X i ) that n Nt = L Hi · X i . i=l

and 0

Corollary 1. As in Theorem 43, let lF be the completed natural filtration of an n-dimensional Brownian motion. Then every local martingale M for lF is

continuous.

Proof

In the proof of Theorem 43 we saw that the underlying probability law P is extremal for A = {X\ . . . , Xn }. Therefore by Theorem 40 ( c ) , ev­ ery uniformly integrable martingale is continuous. The corollary follows by stopping. 0

Corollary 2. Let X = (X 1 , . . . , xn ) be an n-dimensional Brownian motion and let lF be its completed natural filtration. Then every local martingale M for lF has a representation

where H i 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 = (X 1 , . . . , xn ) be an n-dimensional Brownian motion and let lF b e its completed natural filtration. Let Z E F= be in L 1 . Then

there exist

Hi

L(Xi ) with J0= ( H!)2 ds < oo a.s. such that n r= H!dX!. Z = E{Z} + � J i= l 0o

predictable in

188

IV General Stochastic Integration and Local Times

Proof. Let Zt = E{ZI.Ft} , 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] , for some Brownian motion B, 0 < t < 00 . Letting t tend to oo shows that Zoo = Blim,�oo[Z,Z] . , which shows that [Z, Z] 00 < oo a.s. However, a.s. Finally, take t = oo, observe that Zoo = Z and that Zo is constant and therefore Zo = E{Z}.

Fo

is a.s. trivial. Hence 0

X = (X 1 , . . . , xn ) be an n-dimensional Brownian motion and let lF be its completed natural filtration. Let Z E L 1 (.F00 ) and Z > 0 a.s. 00 Then there exist Ji predictable with J0 ( J!) 2 ds < oo a.s. such that Corollary 4. Let

Proof By Corollary 3 there exist predictable H i such that if Zt = then

E{ZIFt} ,

Therefore

n rt 1 . . 1 n t 1 . 2 log ( Zt ) = log ( Zo ) + L J z- H� dX� - L J (H 2 o0 zs2 � ) d . o0 s

i= l

i=l

s

is continuous and never 0, i is locally bounded. ) The proof is completed by setting = i H! and taking exponentials of both s 0 sides.

( Note that since

(Zt)t> O

J!

Corollary 5. Let lF be the completed natural filtration of an n-dimensional

Brownian motion. If a.s.

T is

a totally inaccessible stopping time, then

Proof This is merely Theorem 40 ( a) .

T = oo 0

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) algebra

F.

0

which generate the

a­

Let ( 0, :F, IF, P) be a filtered 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 , M2 , . . . ) in L2 such that they are orthogonal, I: i>- l E{[Mi , Mi l= } < oo, and such that if N is any L2 martingale, then there exists a sequence of predictable pro­ cesses (H 1 , H2 , . . . ) such that Nt = I: i > l J; H!dM;. That is, the martingales { Mi , i > 1} have martingale representation. Proof. Since :F is separable, there exists a countable basis of L 2 , call it (M0 ,M 1 ,M2 , . . . ) , where M0 is constant and E{Mi } 0 for i -/= 0. One can

Theorem 44.

=

i j E{M M } Mi 2 -i

=0 take such a basis to be orthonormal in the usual way. That is, by for i -/= j, and = 1 for all i. If one then multiplies and = we have the construction. The space defines the martingales where the sum converges of stochastic integrals with respect to all of the in L 2 is easily seen to be a stable subspace. Next let be any martingale in L 2 with = 0. Then we have

E{(M i ) 2Mf} E{Mi :Ft}, i

No

Mi

N

where L is orthogonal to the stable subspace generated by the countable This implies that collection of martingales is orthogonal to all = Since these random variables form a basis of the random variables = 0 we have that L is of L 2 , this implies that L must be constant. Since D identically zero.

i ) i> l · (M M/x, Mi .

L= L0

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 =

Fr Fr - . 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{X I :Frn } = E{X I Fr - }. If M is a uniformly integrable martingale, then E{.6.Mr i Fr- }

.6.Mr Fr-

= 0 a.s. and if Therefore is 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 lF is quasi left continuous then martingales jump only at totally inaccessible

stopping times.

A = (At)t >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,, > 0 {At > s + c}, we have that t f---> Tt is right continuous. It is continuous if A has strictly increasing paths. Moreover Ar, > A7, _ > t, and A(r._) - < A(r.) - < t. (Here we use the convention To - = 0.) Finally note that { Ts - < t} = { s < At}, which implies that Tt is a stopping time, and since Tt = lim"_,0 T t E:) we conclude that Tt Definition. Let

-

(+ -

is also a stopping time.

Let a be a positive, finite, right continuous, increasing function on [0, oo). Let c denote its right continuous inverse (change of time). Let f be a positive Borel function on [0, oo) . If G is any positive, finite, right continuous function on [0, oo) with G(O-) = 0, then Theorem 45 (Lebesgue's Change of Time Formula) .

00 00 1 f(s)dG(a(s)) = 1 f(c(s-) )l {c( s - )
and in particular

00 00 1 f(s)da(s) = 1 f(c(s))l{c(s)
Here the integmls are taken over the set [0, ) Proof. First consider f of the form f(s) = l [o ,uj(s), for 0 < u < oo. The left side of the equation then reduces to G(a(u)). For the right side note that fis(c(s-))l{ c( s - )
rem.

0

3 Martingale Representation

191

Corollary. Let a be a positive, finite, continuous, strictly increasing function

on [0, oo) . Let c denote its continuous inverse (change of time) . Let positive Borel function on [0, oo). Then

f

be a

1e(t) f(s)da(s) it f(c(s))ds.

Proof

0

=

0

It suffices to rewrite the left side of the equation as

1= l[o ,e(t)J (s) f(s)da( s )

and observe that c is also continuous and strictly increasing, which implies 1 [0 , t)J (c(s)) = l [o , tj (s). 0 e 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, lF, 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 Me denote the orthogo­ nal projection of M onto S. Then Me is called the continuous martingale part of M. If we write M = Me + (M - Me) as its orthogonal decomposition, then we have Md = (M - Me) where Md 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, lF, 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, M)t � dt .

Let (0, F, lF, 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 an - adapted counting process and let N be its compensator, so that X = N - N is a locally square integrable local martingale. Since N is continuous we have [X, X]t = L s
12

See Exercise 6.

192

IV General Stochastic Integration and Local Times

N

-

{X, X),

and since when jumps at s. Therefore N = continuous paths by hypothesis, we are done.

{X, X) has absolutely 0

Remark (Yan Zeng). With a little more work, one can show the following:

:F,

Let (0, lF, 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 an
t l{t> T},

t If we take the uninteresting counting process Nt = I: i > l l {t > i } which jumps at then its compensator is - constant (and hence predictable) times, simply Nt = Nt, and the martingale X = N -_!{ is identically zero. To avoid these _!;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:

Let (O, F, lF, P) be a filtered probability space satisfying the usual hypotheses. Assume that the a-algebra :F = :F= and that it is separable. 1 3 Assume further that lF 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 = Fr, , 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> l be the countable sequence constructed in Theorem 44. Define the strictly increasing process At = t + I: i> I fMi , Mi ]t, and note that E{ At} < oo for 0 < t < oo. Let C denote the compensator of A. One can check that Ct = t + I:i> 1 {M i ,M i ) t. Then C is both strictly increasing and continuous, since lF is quasi left continuous. Let Tt = inf{s > 0 : C8 > t} and then Ct = inf{s > 0 : T8 > t}, Theorem 47 (Absolutely Continuous Compensators) .

T

where is the time change of the theorem statement. be a G adapted counting process and let Let = As­ sume, by stopping if necessary, that We write G to de­ E note the sharp bracket taken using the G filtration. Note that = Nc, is also a counting process, and since continuous implies that Qc, C (see Exercise 38 of Chap. II) , we conclude is adapted to lF. We then have = = hence = But also = N and we have

N

X= L2 . T L [L, L]t I: s
1 3 A separable a-algebra is defined on page 189.

X N - N. {X, X) Lt Ft [X, X]t t,

4

Martingale Duality and Jacod-Yor Theorem

1 93

{X, X)g, = {L, L)f. Lt - Lt Li> l J; H!dM!

and by the uniqueness of the compensator of L we have we know that On the other hand, by Theorem = for some predictable processes and thus

44i (H ),

for some (predictable) nonnegative processes ci whose existence is assured « because a.s. The process J8 = by Fubini's Theorem. Next, note that since )r = we have (L - L, L - L)�, = Jd"' which by the corollary of Lebesgue's change of time formula, implies = J� Jrsds. 0

d{M i , Mi ) t dCt Li> l (H!)2c�, {L-L, L L = {X, X)g,, {X, X)f J8dC8, {X, X)f -

4 Martingale Duality and the J acod-Yor Theorem on

Martingale Representation

'H2

We have already defined the space 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.

M be a local martingale with cadlag paths. We define the JiP norm of M to be J I M li P = E{[M, M] � 2 P IP . If I M I ·w < we say that M is in JiP. Finally we call JiP the space of all cadlag martingales with finite Definition. Let

oo,

JiP norm.

It is simple to verify that II 111-lv is a norm if we identify martingales that are almost surely equal. Indeed, the only property that is not obvi­ ous is the subadditivity. For the case p = for example, we have that + < + since by the Kunita-Watanabe + < inequality which gives the subadditivity. It is also simple to verify that JiP is a linear space. We state without proof the important martingale inequalities known as the Burkholder-Davis-Gundy in­ equalities, and often referred to by the acronym BDG. (For a proof one can consult many different sources, including and ·

1 11 2 [M N,M Nj ll2 [M,M]1 2112 [N1 ,N] [M,N] [M,M] 1 [N,N] 12 ,

[44].) Theorem 48 (Burkholder-Davis-Gundy Inequalities) . Let M be a mar­ Then tingale with cadlag paths and let p > 1 be fixed. Let Mt = SUPs
Cp

for all t, 0 < t < oo. The constants are universal: they do not depend on the choice of M.

194

IV General Stochastic Integration and Local Times

We have the following simple results concerning

'H 1 .

1 and local martingales of integrable variation are a 2 'H 'H subset of 'H 1 . Proof. First note that if M is a cadlag martingale, then E{ .j[M,M] oo } < 112 which gives the first statement. For the second, if M has (E{[ M , M]oo}) integrable variation then [M,M] oo = L8 (.6.Ms) 2 < (L8 I .6. Ms l ) 2 , whence J[M, M] oo < ]000 l dM81, and taking expectations, we have the result. Theorem 50. 'H2 is dense in 'H 1 and bounded martingales are dense in 'H 1 . Proof. Let M be a local martingale in 'H 1 . By the Fundamental Theorem of Local Martingales (Theorem 25 of Chap. III) we have M = N + U where is a local martingale with jumps bounded by 1 and U has paths of locally Nintegrable variation. If (Tn ) n> l is a sequence of stopping times increasing to a.s., then 1 / MTn / ·r£1 < 1 / M / 1 ?-t l , and also MTn converges to M in 'H 1 . Choose Tn to be the first time [U, U]t has total variation larger than n, and also I [N, Nj Tn / < n - 1; that is, Tn = inf{ t > 0 J� [U, U]t > n, or [N, N] t > n - 1} with Tn increasing to Then [N, N] Tn < n and thus NTn is in 'H 1 for each n. Therefore we turn out attention to urn , and since [U, U] Tn < n + / .6. Urn I , we need consider only the one jump of U at time Tn .6.Urn . Let � denote .6.Urn . Letting �k = l;1 {1 � 1 < k} > since �k is bounded we can compensate �k 1 {t > Tn } > and call the resulting martingale Vt. The compensator of �kl{t >Tn } is in 'H2 because �k is bounded. We have then that v; is in 'H2 and converges to Vn as k in total variation norm, and hence also in 'H 1 . This proves that 'H 2 is dense in 'H 1 , and since bounded martingales are dense in 'H 2 , they too are dense in 'H 1 . Theorem 51. Let M be a local martingale. Then M is locally in 'H 1 . Proof. By the Fundamental Theorem of Local Martingales we know that M = + U, where N has jumps bounded by a constant /3 and U is locally of Nintegrable 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) . c

Theorem 49.

0

oo

:

oo .

:

---> oo

0

0

One can further show, by identifying JiP with LP through the terminal value of the martingale, that JiP is a Banach space (in particular it is complete) for each p > One can further show that the dual space of continuous linear p oo. This continues nicely functional on JiP is where � + ! = It can be shown (see the analogy with LP except for the case p = is not that the dual of It turns out that a better analogy than LP is that of Hardy spaces in complex analysis, where C. Fefferman showed that can be identified with the space of functions of bounded mean the dual of oscillation, known by its acronym as With this in mind we define the space of martingales.

1.

'H 1 BMO

'H 1

'Hq ,

'H00

1, 1 < < 1.

.

BMO.

[47] )

4 M art ingale Duality and Jacod-Yor Theorem

1 95

Definition. Let M be a local martingale. M is said to be in BMO if M is

'H2

in and if there exists a constant have

c

such that for any stopping time

E{(Moo - Mr - ) 2 1 Fr} < c2

T we

a.s.,

where M0 _ = 0 by convention. The smallest such c is defined to be the BMO norm of M, and it is written IIM II BMO · If the constant c does not exist, or if M is not in then we set II MIIB MO = oo.

'H2,

c2

Note that in the above definition E {M! I Fo } < (with the convention that Mo- = 0) and therefore IIMII'H2 < II MII BMO· Note in particular that , II M II BM O = 0 implies that M = 0. 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 the BMO norm:

T. This in turn gives us an equivalent description of

- Mr - )2} E{(Moo I IM II BMO = sup P(T < oo) where the supremum is taken over all stopping times T. Note that this second T

characterization gives that II · IIBMO is a semi-norm, since it is the supremum of quadratic semi-norms. An elementary property of BMO martingales is that M E BMO if and only if all of the jumps of M are uniformly bounded. Thus trivially, continuous L2 martingales and bounded martingales are in BMO . The next inequality is quite powerful.

Let M and N be two local mar­ tingales. Then there exists a constant c such that

Theorem 52 (Fefferman's Inequality) .

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 = f; u;d[M, M] s and define H and K by

2

Ht =

y'lJ; +

u? -/Ct l

#t 2 = t. C K >O}, l{ t C t {t > O}

IV General Stochastic Integration and Local Times

196

Using integration by parts yields

Ht2 d[M, M] t = l{ c,>o } v'(J; dCt t + y'Ct - 1 {t>O} From the definitions of H and K we have Ht2 Kt2 > 21 ut2 1 { c, >o} ·

=

1 {t>o} d Vr;::y Ct .

The Kunita-Watanabe inequality implies

1

00

00

/Us/1 { cs =o } / d[M, N] s / < ( 1 u;1 {Cs =o } d[M, M] s ) 1 12 ([N, N]oo ) 1 12 00 = ( 1 1 { Cs=O} dC8 ) 1 12 ([N, N] 00 ) 1/ 2 = 0 a.s.,

/

and since /d[M, N] 8 is absolutely continuous with respect to consequence of the Kunita-Watanabe inequality, we have

d[M, M] s as a

oo oo 1 r r y12 E{ j /Us l / d[M, N] s/ } = yl2 E{ j / 1{ Cs >o} Us l / d[M, N]s /} 0 00 0 < E{ 1 / HsKs/ ld[M, N]s/} 00 00 < E{ 1 Hid[M, M]s} E{ 1 Kid[N, N]s} · 1

,.-----

But

00

00

E{ 1 H;d[M, M]s} < E{ 1 d �} = E{ JC:} 00 = E {( 1 U]d[M, M] s) 1/ 2 } and

oo

oo

E{ 1 K; d[N, N] s } = E{ 1 ([N, N]oo - [N, N]s - )dK;} 00 = E{ 1 (E{[N, N]oo/Fs } - [N, N] s - )dK;}. But E{E{[N, N]oo / Fs} - [N, N]s- } is bounded by we have that

00

//N I/ �MO on (0, oo), hence

00

E{ 1 K; d[N, N]s} = E{ 1 (E{[N, N]oo / Fs} - [N, N]s - )dK; } < / N / I � M E{ JC:}, o

and the result follows.

0

4 Martingale Duality and Jacod-Yor Theorem

Remark. The constant c in Theorems

197

52 and 53 can be taken to be J2, as

can be seen from an analysis of the preceding proof.

Let N E H 2 • Then N is in BMO if and only if there is a constant c > 0 such that for all M E H2 , jE{[M, N]oo}l < c jj M II w · Moreover IINII B MO < V6 c. Proof If N is in BMO, then we can take c = V2 11N II B MO from Fefferman ' s inequality (Theorem 52) and the remark following the proof of Theorem 53. Now suppose that M is in H2 and that jE{[M, N]oo} l < cjjMII w ; we want to show that N is in BMO. We do this by first showing that I No l < c a.s. , and then showing that N has bounded jumps. Let A = {INol > c} . Suppose P(A) > 0. Let � = si�/�o) 1A. Then E{l �l} = 1 , and if we define the trivial martingale Mt = � for all t > 0, then M E }{2 and I I M I Iw = E{l�l} 1, whence 1 { l ol > c jN j E{[M, N]oo}l = E{MoNo} = E{ P N0 > c } > c = c i! M IIw · This of course is a contradiction and we conclude jNo l < c a.s. We next show j�Nj < 2c. Since every stopping time T can be decomposed Theorem 54.

�i � ;

=

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(j�Nrl > 2c) > 0, and set sign( �Nr)

� = P(j�Nr l > 2c) l{ I�Nr l > 2c } · Let M be the martingale consisting of � 1 {t >T } minus its compensator. Then M is in H2 and has at most one jump, which occurs at T. The jump is given

by

�MT =

{ � - E{�IFr- },

T is predictable, T is totally inaccessible.

� Note that we also have II M II w 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 1 { 1 �Nrl > 2c} �N r j j t: E{[M ' N] oo } = E{�MT�NT } = E{ �NT } = E{ P( �N 2c) } j rj > > 2c > c IIMII w , '>

which is a contradiction. On the other hand, if T is predictable, then we know that E{�Nr iFr - } = 0 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([.6.Nr[ > 2c) = 0, and thus N has jumps bounded by 2c. Last let T be any stopping time. Let M = N - NT and TJ = [N, N] 00 [N, N] r - Then M is in 7t2 and [M, M] 00 = [M, N]oo = TJ · By our hypotheses it now follows that

E{ry} = E{[M, N]oo} < c [[ M [[ rtl = cE{ �} = c E{J77 l{ T
E{[N, N] oo - [N, N] r} = E{ry} < c2 P(T < oo). This then implies

E{[N, N] oo[Fr} - [N, N] r < c2

a.s. ,

and since almost surely, we obtain which yields the result.

D

During the proof of Theorem 54 we proved, inter alia, that the jumps of a local martingale in BMO are bounded. We formalize this result as a corollary, which in turn has its own corollary.

Corollary 1. Let

jumps.

Corollary 2. Let

bounded.

N be a local martingale in BMO. N

be a local martingale in

Then

BMO.

N has bounded

Then

N

is locally

Let c be a bound for the jumps of N which we know exists by the previous corollary. Next let Tn = inf{t > 0 : [Nt[ > n }. Then [NTn [ < n + c, and N is locally bounded. D

Proof

The key result concerning H 1 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 L N from H 1 to lR by

L N (M) = E{[M, N]oo}

for all M in 1t 1 . Then one can easily check to see that L N 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 H 1 then it is also complete, a fact that is apparently not easy to verify directly.

4 Martingale Duality and Jacod-Yor Theorem }[1 is r£l

1 99

BMO).

The Banach space dual of all (bounded) linear junctionals on can be identified with B M0 . Moreover if LN is such a functional then the norms II LN II and II N IIBMO are equivalent. Proof. Let N be in BMO. By Fefferman 's inequality we have

Theorem 55 (The Dual of

M in }[1 . This shows that LN is in the dual of }[1 and also that Il LN il < c II N IIBMO · Note further that LN cannot be trivial since LN (N) = LN > 0 unless is identically 0. Therefore the mapping is an injective linear mapping from BM 0 into Hh, the dual of }{1 . Let be an arbitrary linear functional in the dual of H 1 . We have I L(M) I < II L II II M II rtl < II L II II M II w · This means that L is also a bounded linear functional on H2. Since H2 is isomorphic as a Hilbert space to the L2 space of the terminal random variables of the martingales in }{2, we have that there must exist a unique martingale in }{2 such that for any M in }{1 we have: for all

E{[N, N] oo } L

N

F(N)

=

N

Clearly LN = L on }{2 , and since }{2 is dense in H1 by Theorem 50, we have that and L N are the same functional on }{1 . This shows that B M0 equipped with the norm II LN II is isomorphic to }{1* 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 I I LN II and II N II BMO are equivalent norms. This completes the proof. D

L

While Fefferman's inequality, the space of BMO martingales, and the du­ ality of H00 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.

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) = 0 is the functional which is identically zero.

Theorem 56 (Hahn-Banach Theorem) .

Theorem 57 (Jacod-Yor Theorem on Martingale Representation) .

Let A be a subset of H2 containing constant martingales. Then S(A), the stable subspace of stochastic integrals generated by A, equals H2 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.

IV General Stochastic Integration and Local Times

200

Proof

The necessity has already been proved in Theorem 38. By the Hahn­ Banach Theorem, r£1 = 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)) = 0. Then there exists a martingale such that L = LN . The local martingale N is locally bounded, N in so by stopping we can assume it is bounded and that N0 = 0. ( See the two corollaries of Theorem 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

BMO

54.)

Noo Noo dQ = (1 - 2c )dP, dR = (1 + 2c )dP. R are both in M 2 (A), and P = �Q + �R

Then Q and shows that P not extremal in M 2 (A) , a contradiction. Therefore we must have that L identically zero and we have that r£1 = S(A) . As far as H2 is concerned, is a subspace of }{ 1 , hence }{ 2 C S(A) . But by construction of S(A) , it contained in H 2 , and we have martingale representation.

is is it is D

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 H we have both that X0 = 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 Y are 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 dtdlag 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 Y are 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) . ¢ : JR. IR be Lipschitz continuous. Then Emery 's structure equation

Let

---+

has a weak solution with both (Xt ) t >o and (J� rj;(Xs - )dXs) t >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.

Let X be a (weak) solution of (* * *). Then the following hold. (i) E{Xf} = 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 f:l.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

Theorem 59.

-

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

(T1 n ) n> l L.

of stopping times increasing to oo such that there exists a sequence J;I\Tn ¢(X8_)dX8 is in Therefore E{[X, X]t/\ Tn } = E{t i\ } , and apply­ ing the Monotone Convergence Theorem to each side of the equality in this equation yields E{[X, X]t} = t which further implies that X is a martingale on [0, t] for each t < oo and that E{Xf} = E{ [X, X]t} = t. For the second statement, recall that

Tn

[X, X]t = [X, X]f + �) �Xs) 2 , s
and hence we have 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; ¢(X8_)dX8• 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 = 1\ TB where A is the accessible part of and B is the totally inaccessible part of Since can be covered by a countable sequence of predictable times with disjoint graphs, we can assume is predictable without loss of generality. Thus it suffices to show P(A) = 0. However since is predictable, and X is a uniformly integrable martingale, we have E{ �XTA I FrA _ } = 0. But �XrA = ¢(XrA - ) by part ( ii) of this theorem, and ¢(XrA - ) E FrA _ since is predictable, which implies that the jump of X at is zero, which in D turn implies that P(A) = 0.

T.

TA

T TA

TA

TA

TA

TA

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

d[X, X]t = dt + (a + f3Xt-)dXt,

f3 = 0, it reduces to d[X, X]t = dt + adXt , and when in addition a = 0 we have seen that X is standard Brownian mo­ tion. Note that Levy's Throrem gives us weak uniqueness in this case, since and when

5 Examples of Martingale Representation

203

any solution with a = f3 = 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 + (a + f3Xt- )dXt, Xo = 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 (n, 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 X0 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 P is 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 + .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 ¢(Xs - )dX8 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 D consists of only one, and thus extremality is trivial.

J�

(1 -

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 = In this case the equation of Theorem with X0 = becomes, in integral form,

60

0.

0

[X, X]t = t + (Xt - Xo) = t + Xt. Therefore Xt = [X, X]t - t and hence X is a finite variation martingale. Moreover f1Xt = 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 Xt = 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-3; "' - -; a

is the unique solution of the equation of Theorem 60 if N is a standard Poisson process. Note that (as is well known) converges (weakly) to Brownian motion as a 0. We now consider the more interesting cases where f3 of- 0. We repeat the equation of Theorem 60 here (in integrated form) for ease of reference:

xo:

--+

Observe that solution of

X is a solution of the above equation if and only if X + � is a (00)

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

with initial condition

where M is Azema's martingale15, B is standard Brownian motion, and 9t = sup{s < t : B8 = By Theorem 86 of Sect. 8 of this chapter, we know that [X, X]f = and [X, X]t = 2gt. Integration by parts gives J; X8_ dX8 = (t - gt) - 9t = t - [X, X]t, since [X, X]t = Ls
0

15

0 }.

Azema s martingale is treated in detail in Sect. '

8 later in this chapter.

205 M is 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 t [X, X]t - t = -2 1 Xs_dXs 6 Stochastic Integration Depending on a Parameter

=tXfbecause - [X, X]t. Equat­ = 0 we

and using integration by parts we obtain 2 J� X8_dX8 ing terms gives x; t, and since E{Xt} 0 for all deduce that

= P(Xt = vt) = P(Xt = -vt) = 21

=

Xo

-

for all t > 0. The jumps of X occur from a change of sign, and they arrive according to a Poisson process with intensity lt dt. 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,

N

is independent of the O"-algebra Fs and has a Poisson distribution where with parameter .A ! ln( �). 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 [2 1 ] and Jacod [103] . Throughout this section (A, A) denotes a measurable space.

9

Theorem 62. Let y n ( a, t, w ) be a sequence of processes that are (i) A 0 B(IR + ) 0 F measurable, and (ii) for each fixed a the process yn (a, t,w) is cadlag. Suppose yn (a, t, ) converges in ucp for each a E A. Then there exists an A 0 B(!R+ ) 0 F measurable process Y Y(a, t, w ) such that

=

·

(a) Y(a, t, ) = limn_,00 yn (a , t, ) with convergence in ucp; (b) for each a E A, Y is a.s. cadlag. Moreover there exists a subsequence nk (a) depending measurably on a such that limnk( a) -> oo �nk (a) = yt uniformly in t on compacts, a.s. ·

·

206

IV General Stochastic Integration and Local Times

I

Let s� ,i , j = SUPt < u Yi (a, t, ·) - Y1 (a, t, ·)1. Since yi is cadlag in t the function (a, w) 1---t S�, i ,J is A 0 F measurable. By hypothesis we have lim i ,J ->oo S� i J = in probability. Let no( a) = 1 , and define inductively ,,

Proof.

0

nk (a) = inf{m > max(k, nk - 1 (a)) : sup P(Sf:'i 1 > T k ) < T k }. i ,j > m '

We then define Since each

zk (a ' t ' w)

= yn a (a t w) . k( )

' '

a 1---t nk (a) is measurable, so also is zk . Define 1 r::'i ' = sup iZi ( a, t, w) - z (a, t, w)i; t u 1·

<

then also (a, w) I-t r::, i , j (w) is jointly measurable, since zi have cadlag paths (in t). Moreover by our construction P(Tf:, + m > 2- k ) < 2- k for any m > 1. k,k The Borel-Cantelli Lemma then implies that limi,J ->oo r:;, i J = 0 almost surely, ,, which in turn implies that lim zi (a, t, ) exists a.s., t -> 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 ( a, t, w ) =

{

limi ---->00 zi (a, t , w), 0,

Then Y is cadlag thanks to the uniform convergence, and it is jointly mea­ surable. D

Theorem 63. Let X be a semimartingale with X0 = 0 a.s. and let H(a, t, w) = H't (w) be A 0 P measurable1 6 and bounded. Then there is a function Z (a, t, w ) in A 0 B(JR + ) 0 F such that for each a E A, Z(a, t, w) is a cadlag, adapted

version of the stochastic integral J; H�dX8• Proof. Let 1t = {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

lo

t

H(a, s, ·)dX8

= lot f(a)K(s, ·)dX8 = f(a) lo

t

K(s, ·)dX8,

and thus clearly H = fK is in H. Also note that 1t is trivially a vector space, and that of the form H = fK generate bA 0 P. Next let Hn E 1t and suppose that Hn 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. D

H

16 Recall that P denotes the predictable u-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(JR+ ) 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 Ha l{ I Ha i < k} "X 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 semimartingale, 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(JR + ) 0 :F measurable such that for each a, za is a cadlag version of Ha · X. Then yt f Zf'J.l (da) is a cadlag version of H · X, where Ht f HfJ.l(da) . Proof. By pre-stopping we may assume without loss of generality that X E =

=

=

=

=

A

A

rf.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} < oo. 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 l f(a)IJ.l (da) < oo . In this case we have Zf = f(a)K · X, and moreover

j Zff.l (da) j f(a)K · Xf.l (da) =

= = =

K · X j f(a)J.l(da)

(! 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 Hn 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, .....

� L

(E{ ll ll

s�p IZ� ,t

< E{ r sup IZ� t

- Zfl f.l (da)} ) 2 - Zfi 2 J.l (da)}

r E{sup IZ� t - Zfi 2 }J.l(da) '

t } } t < 4 E{(Z� , oo - Z� ) 2 }J.l(da) 4 E{[Z� - z a , z� - za ] oo }f.l (da) by Doob ' s quadratic inequality for the martingales Z� and za , and by Corol­ lary 3 of Theorem 2 7 of Chap. II. Continuing, the preceding equals

L

A

'

=

=

A

L

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 < a.s. t A } and therefore fA IZflp,(da) < oo for all t, a.s. Moreover

r sup IZ� ,t - Z�lp,(da)

()()

r Zfp,(da) l} < E{ r sup IZ�,t - Zflp,(da)} fl(da) z� t t }A ' }A }A t which tends to 0. Therefore taking Hn ,t = f H� ,t p,(da) we have Hn · Xt = fA Z�, t fl(da) converges in ucp to f Zfp,(da). Since Hn · X converges to H · X by Theorem 32, we conclude H · X = J Zfp,(da). 0 E {sup I r

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) . martingale, let Hf = H (a, t, w) be A 0 P measurable, let p,

measure on A, and assume

Let X be a semi­ be a finite positive

Letting Zf = f� H:dxs be A 0 B(lR+ ) ® F measurable and za cadlag for each a, then yt = fA Zfp,(da) exists and is a cadlag version of H · X, where Ht = fA Hfp,(da) . Proof. By pre-stopping we may assume without loss of generality that X E 7t2 and that II Ha I I L 2 ( dl-' ) is ( 7t2 , X) integrable. Let X = N + A be the canonical decomposition of X. Then Next observe that

00

00

E{ 1 IIH� II L 2 (dl-' ) l dAsl } > E{ll 1 IH� I IdA siii L2 ( d!-') } oo > cE{ll fo I H�II dAsiii L 1 (di-L) } 00 = c E{ 1 IH� I IdAsl }p,(da). Also

L

E{1

00

6

Stochastic Integration Depending on a Parameter

L (H� )2 jj(da)d[N, N] s} = L E{1

00

209

(H�) 2 d[N, N]s}f.l ( da),

and therefore E{j000 I H� I I d As l} < oo and E{j000 (H�)2d[N, N]8} < oo for jj-almost all a E A. Whence Ha E L(X) for Jl almost all a E A. Next define Hn = H1 { IH I< n } ' and the proof of Theorem 64 works as well 0 h�. The hypotheses of Theorem 65 are slightly unnatural, since they are not invariant under the transformation

where i.p is any positive function such that f �.p(a)Jl ( da) < oo . This can be alleviated by replacing the assumption ( fA (Ha )2jj(da)) 1 12 E L(X) with a)2 ( H (j cp (a) Jl (da)) 1 12 E L(X) for some positive i.p E L 1 (djj). One can also relax the assumption on Jl to be a-finite rather than finite.

Example. The hypotheses of Theorem

65

are a bit strange, however they are in some sense best possible. We give here an example of a parameterized process (Ha ) a E A , a positive, finite measure Jl on A, and a semimartingale X such that

(i ) (a, t) Hf is A 0 P measurable, (ii ) Ha E L(X), each a E A, and ( iii ) fA I Hf lf-l(da) E L(X) -+

but such that if Zf = f; H�dX8 then fA Zfjj(da) does not exist as a Lebesgue integral. Thus a straightforward extension of the classical Fubini Theorem for Lebesgue integration does not hold.

Indeed, let A = N = { 1 , 2, 3 , . . . } and let Jl be any finite positive measure on A such that /-l( {a}) > 0 for all a E A. Let X be standard Brownian motion, let t0 < t1 < t 2 < be an increasing sequence in [0, 1], and define · · ·

Then

is in

L2(dt), whence Ht = fA IHflf-l ( da) E L(X) , and moreover if t > 1, 00 1 H Xt = """ 0 - (ta - t a - t) 1/ 2 (Xta - Xt a-I ) , a=l a ·

where the sum converges in

L2 • However if t > 1 then

210

IV General Stochastic Integration and Local Times

and

because (ta - ta - t) -1 12 (Xta - Xt ) is an i.i.d. sequence and 2::;:' 1 a - 1 = oo . Note that this example can be modified to show that we also cannot replace a

the assumption that

_

,

(L (H:)2Jl(da)) 112 E L(X)

with the weaker assumption that

(jA ( Hf)Pjj(da)) 1 1P E L(X) for some p < 2.

7 Local Times

32

In Chap. II we established Ito's formula (Theorem 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 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 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) .

66

[34]. )

Let f lR --> IR be convex and let X be a semimartingale. Then f(X) is a semimartingale and one has

Theorem 66.

:

f(Xt) - f(Xo) Jt f'(Xs_)dXs + At o+ where f ' is the left derivative of f and A is an adapted, right continuous, increasing process. Moreover flAt = f(Xt) - f(Xt - ) - f'(Xt_ )LlXt. Proof First suppose lXI is bounded by n, and in rt2, and that X0 0. Let g be a positive coo function with compact support in ( OJ such that 00 00 j 00 g(s)ds 1 . Let fn (t) = n j 00 f(t + s)g(ns)ds. Then fn is convex and C 2 and moreover f� increases to f' as n tends to oo. By Ito's formula =

=

-oo,

=

7 Local Times where

�1

211

t

A� = L {fn (Xs) - fn (Xs - ) - f�(Xs_)boXs} + J;: (Xs - )d[X, X]� . 0 O<s
to

oo ,

n

we obtain

where lim n oo Af = 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 J� f'(X8_)dX8 = 0 we have that A0 = 0. When t > 0, the jump of the left side of ( * ) is f(Xt) - J(Xt-) , while the jump of the right side equals f'(Xt_)boXt + boAt. Therefore boAt = J(Xt) - J(Xt- ) - f'(Xt_)boXt, and the theorem is established for lX I bounded by n and in rf.2 . Now let X be an arbitrary semimartingale with X0 = 0. By Theorem 13 we know there exists a sequence of stopping times ( Tn ) n >l , increasing to oo a.s. such that xrn - E rf.2 for each An examination of the proof of Theorem 13 shows that there is no loss of generality in further assuming that I XTn - 1 < also. Then let yn = X 1 ro T ) and we have --->

n.

,

n,

n

which is equivalent to saying

[O, Tn ) . One easily checks that (An+ l ) Tn - = (An ) rn - , and we can define A = An on [0, Tn ), each n. The above extends without difficulty to functions g JR 2 lR of the form g(Xt, H) where H is an F0 measurable random variable and x g(x, y) is convex for every y. For general X we take Xt = Xt - X0, and then j(Xt) = j(Xt + Xo) = g(Xt, Xo) , where g(x, y) = f(x + y). This completes the proof. on

:

---+

�--*

0

Notation. For x a real variable let x+ , x - be the functions x + - max (x, 0) 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

martingales.

_

X be a semimartingale. Then lXI , x+, x-

are all semi­

212

IV General Stochastic Integration and Local Times

The functions f(x) lx l , g(x) = x + and h (x) = x - are all convex, so the result then follows by Theorem 66. 0

Proof

=

Corollary 2. Let semimartingales.

X, Y

,

be semimartingales. Then

XVY

and

X 1\ Y

are

Proof Since semimartingales form a vector space and xVy = � (lx-yl +x+y) and x 1\ y = � ( x + y - lx - y l ) , the result is an immediate consequence of Corollary

0

1.

We can summarize the surprisingly broad stability properties of semi­ martingales.

The space of semimartingales is a vector space, an algebra, a lattice, and is stable under C 2 , 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 C 2 transformations (Theorem 32: Ito's Formula) . Theorem 67.

That they form a lattice is by Corollary 2 above, and that they are stable 0 under convex transformations is Theorem 66.

Definition. The sign function is defined to be sign(x) =

{ -1 ' 1

'

if X > 0, if X < 0.

Note that our definition of sign is not symmetric. We further define •

ho(x) = lxl and ha (x) = lx - al . Then sign(x) is the left derivative of h0 (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 - ai = IXo - ai + t sign(Xs - - a)dX8 + A� , ( * * ) lo+ 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 (X8) - ha (X8_ ) - h� (X8_ ) LlX8 }. O<s
Notice that by the corollary of Theorem 63 the integral J;+ sign(X8_ - a)dX8 in ( * * ) has a version which is jointly measurable in (a, t, w) and cadlag in

7 Local Times

213

t. Therefore so does (Af)t>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:sO is continuous in t. Indeed, the local time L a is the continuous part of the increasing process Aa . The next theorem is quite simple yet crucial to proving the properties of La that justify its name.

Theorem 68.

Then

Let X be a semimartingale and let La be its local time at a.

t (Xt - a) + - (Xo - a) + = l{ X ,_ > a } dXs + L l { X ,_ > a } (Xs - a) 0+ O<s
1

t

10+ l {x,_ a} (Xs - a)+ L l{x ,_ < a } (Xs - a) + + �Lf . O<s
=

=

1O+ J'(Xs_ )dXs + ct t g(Xo) + 1 g' (Xs - )dXs + ct­ O+

j(Xt) = j(Xo) + g(Xt) = Next let

t

Dt = ct - L {f(Xs) - f(Xs - ) - J'(Xs - )�Xs},

Dt; = c; - L {g(Xs) - g(Xs - ) - g'(Xs - )�Xs } O<s
214

IV General Stochastic Integration and Local Times

Let X be a semimartingale, 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 : X8_ (w) = X8(w) = a}. Proof Since Lf has continuous paths, the measure dLf(w) is diffuse, and since { s : X8_ (w) = a} and {s : X8 _ (w) = X8(w) = a} differ by at most a countable set, it will suffice to show that dLf(w) is carried by the set {s : X8_ (w) = a}. Suppose S, T are stopping times and that 0 < S < T such that [S, T) c {(s, w) : X8_ (w) < a} - {X_ < a}. Then X < a on [S, T) as well. Hence by Theorem 69.

the first equation in Theorem

68 we have

(X - a) � - (X - a ) � = J( l{ X. - >a } dXs + L l{ Xs - > a} (Xs - a) s S<s< T + L l{x,_ 0, by

r, if Xr_(w) < a, Sr (w) = oo , if Xr - (w) > a.

{

Then define

Tr(w) = inf {t > Sr(w) : Xt_ (w) > a}. Then [Sr, Tr) C {X_ < a}, and moreover the interior of the set {X_ < a} equals Ur E IQ!, r>O ( Sr, Tr). As we have now seen, dLa does not charge the interior of the set {X_ < a}. This set is open on the left, hence it differs from its interior by an at mo �t countable set, and since dLa is diffuse it doesn't charge countable sets. Thus dLa does not charge the set {X_ < a} itself. Analogously one can show dLa does not charge {X_ > a} . Hence its 0 support is contained in the set {X_ = a}, and we are done. The next theorem gives a very satisfying generalization of Ito's formula (Theorem 32 of Chap. II) .

Let f be the difference of two convex functions, let f' be its left derivative, and let J.l be the signed measure (when restricted to compacts) which is the second derivative of f in the generalized function sense. Then the following equation holds: t f(Xt) - f(Xo ) = 1 f'(Xs_)dXs + L {f(Xs) - f(Xs - ) - f'(Xs_) Ll Xs} O+ O<s
1 100 J.l( da)Lf ' - oo

+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 (i.e. , 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 g by

g(x) =

� j l x - Y iJ-L(dy).

It is well known that f and g 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) = lx - Y i J-L (dy). Then f'(x) = sign ( x - Y ) J-L (dy) and f" ( x) = 1-l ( dx). Moreover if

�J

�J

Jl = L I Xs - Y i - IXs - - Yi - sign(Xs - - y)fl.X8 , O<s
then

� J JlJ-L (dy) = O<s
(1)

� J HlJ-L(dy) = f(Xt) - / (Xo) ­

(2)

Also, letting

Hl = IXt - Y i - IXo - y i , one has

Next consider Zl = J;+ sign (X8_ - y)dX8, and let ZY be the B(IR) ® B(IR + ) ® :F measurable version, which we know exists by Theorem 63. By Fubini's Theorem ( Theorem 64) we have

(3 ) Since

Lf = Hf - Jf - Zf, we have

� j LfJ-L(da) = � j HfJ-L (da) - � j JfJ-L(da) - � j ZfJ-L (da)

(4)

and combining (4) with (1), (2 ) , and (3) , yields the formula ( * ) . It remains to reduce the general case to that of 1-l supported on a compact interval with finite total mass. By linearity it further suffices to treat the case where f is convex. Define

f(n) + f'(n)(x - n), fn (x) = f(x), f(-n) + f'(-n)(x + n),

if x > n, if - n < x < if x < - n.

n,

Then fn is also convex and its second (generalized) derivative J-Ln (dx) agrees with J-L on [-n, n) and is of finite total mass . Let y n = fn (X), and T n =

IV General Stochastic Integration and Local Times

216

inf{t > 0 : IXtl > n}. Note that yn f(X) on [O, T n ), and that Lrn 0 for L�J..t (da) for t < Tn , all a, fa l > n, by Theorem 69. Therefore j L�J..tn (da) and by the preceding ( *) holds for Y fn (X) f (X) on [0, rn ). Therefore (*) holds on [O, Tn ). Since the stopping times (Tn ) n > l increase to oo a.s., we have ( * ) holds on all of IR + X n, and the theorem is proved. 0

=

=

=

=

=f

The next formula gives an interpretation of semimartingale local time as an occupation density relative to the random "clock" d[X, X]�. l 7

Corollary 1. Let

X be a semimartingale with local time ( La ) a EIR· Let g be

a bounded Borel measurable function. Then a.s.

t

J: Lfg(a)da = 1 g(X8_ )d[X, X]� .

Let f be convex and C2 . Comparing (*) of Theorem formula (Theorem 32 of Chap. II) shows that

Proof.

70

with Ito's

100 Lff"(a)da = Jto f"(X8_)d[X, X] � -oo

where J..t (da) is of course f"(a)da. Since the above holds for any continuous and positive function f", 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

Corollary 2. Let

1.

X be a semimartingale with local time (La ) a EIR· Then [X, X]� = 1: Lfda.

Corollary 3 (Meyer-Tanaka Formula) . Let

continuous paths. Then

X be a semimartingale with

I Xtl = I Xo l + ro sign(Xs)dXs + L�. J+

Proof. This is merely Theorem 70 with f(x) = lxl, which implies J..t(da) 2c0(da), point mass at 0. The formula also follows trivially from the definition 0 of L0 .

=

=

If Xt Bt is a standard Brownian motion, then Mt J; sign(B8)dB8 is a continuous local martingale, and [M, M]t sign (B8) 2 d[B , B]8 [B, B]t t. Therefore by Levy's Theorem (Theorem 39 of Chap. II) we know that Mt

= J;

1 7 Recall that [X, xr denotes the path-by-path continuous part of t [X, X]0 0.

=

=

>-->

=

[X, X]t with

7 Local Times

is another Brownian motion. We therefore have, where Brownian motion,

217

B0 = x for standard

J;

where f3t = sign (B8)dB8 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) = lxl, then f" = 28(x) , where 8 is the "delta function at 0," which of course is a generalized function, or "distribution." Thus Corollaries 1 and 3 give the intuitive interpretation of local time as L� = J� 8(X8)d[X, X]8, and Lt = J� 8(X8 - a)d[X, X]8, for continuous semi­ martingales. Local times also allow us to give simple examples of functions f that are only slightly more general than convex functions and are such that f(X) need not be a semimartingale when X is one. Let f(x) = lxl a , 0 < a < 1 . Then f cannot be expressed as the difference of two convex functions because the slope of f becomes vertical as x decreases to zero. The proof of this result is simpler if we restrict a to 0 < a < 1/2 .

Let X be a continuous local martingale with X0 = 0, and let 0 < a < 1/2. Then yt = IXt I a is not a semimartingale unless X is identically zero. Proof Let us suppose that Y = IXI a is a semimartingale. Let f3 1 /a > 2. Theorem 71.

=

We then have

t

t ( 1) IXt I = ypt f3 10 y8/3 - l d Y. + /3 /32 10 y8/3 - 2 d[Y Y] =

S

l

Sl

since Y has continuous paths. Using Theorem 69 and the Meyer-Tanaka for­ mula ( Corollary 3 of Theorem 70 ) we have, where Lt = L� is the local time of X at 0,

t

Lt = 1 1 { Xs=O} dLs t t = - 1 1{xs =O} sign (X8)dX8 + 1 1 {xs=o} d 1 XIs

yn

is a sequence The integral J; 1 { Xs =O} sign (X8)dX8 is identically zero. For if n of stopping times increasing to 00 a.s. such that xr is a bounded martingale, then

E{(

1

t/\Tn

1 { X s=O} sign (X8)dX8 ) 2 } E{ 1 =

t /\Tn

1 { Xs=O} d[X, X]8}

= E{1: Lf/\Tn 1 {o} (a)da} = 0.

218

IV General Stochastic Integration and Local Times

Therefore Lt = L� = becomes

J� 1{x.=o} djXj8• Since {Xs = 0} equals {Ys = 0}, this

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 0. Such a local martingale must be identically zero. This completes the ��

0

It is worth noting that if X is a bounded, continuous martingale with X0 = 0, then yt = !Xt!", 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 [0, l] n in R.n , and let �m denote all x E � whose coordinates are of the form k2-m, 0 < k < 2m.

Let (E, d) be a complete metric space, and let ux be an E-valued random variable for all x dyadic rationals in R.n . Suppose that for all x, y, we have d(Ux , UY ) is a random variable and that there exist strictly positive constants C, f3 such that

Theorem 72 (Kolmogorov's Lemma) .

E,

Then for almost all w the function x ...... u x can be extended uniquely to a continuous function from R.n 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 !Rn to the reader. Two points x and y in Ll m are neighbors if supi j xi yi j = 2- m . We use Chebyshev's inequality on the inequality hypothesized to get P{ d(U x , UY) > T "m } < C2""'m T m (n+t3) . Let

Am =

{w :

Since each x E 2 mn we have

:3

neighbors x, y E

Llm

'

Llm with

d(Ux (w), UY(w)) > T"m }.

has at most 3n neighbors, and the cardinality of Llm is

P Am ) < c2m (ac - t3)

(

c = 3n c. Take a sufficiently small so that ac < {3. Then P(Am ) < cT m 8 where (j = {3-ac > 0. The Borel-Cantelli Lemma then implies P(Am infinitely often) = 0. That is, there exists an m0 such that for m > mo and every pair u, v ) of points of Llm that are neighbors,

where the constant

(

ux

We now use the preceding to show that x f--+ is uniformly continuous on Ll and hence extendable uniquely to a continuous function on [0, l] n . To this end, let x, y E Ll be such that ll x - Yll < 2- k - I . We will show that < c2-ak for a constant c, and this will complete the proof. Without loss of generality assume k > m0 . Then x = (xl , . . , xn and y = (y 1 , . . . , yn ) in Ll with ll x - Y ll < 2- k -1 have dyadic expansions of the form

d(Ux , UY )

.

)

ui + 2:= ajT 1 j>k y i = v i + 'L b; TJ j> k where aj, bj are each 0 or 1, and u, v are points of Ll k which are either equal or neighbors. Next set uo = u, u1 = uo + ak+ 12- k - 1 , u2 = u1 + ak+ 22- k - 2 , . . . . We also make analogous definitions for vo, v1, v2, . . . . Then ui - 1 and Ui are equal or neighbors in Ll k+i> each i, and analogously for vi_1 and vi. Hence xi

=

d(Ux (w), uu (w)) < Lk T"J j= d(UY(w), Uv (w)) < 2:==k T"J 00

00

j

220

IV General Stochastic Integration and Local Times

and moreover D

The result now follows by the triangle inequality.

Comment. If the complete metric space (E, d) in Theorem 72 is separable, then the hypothesis that d(Ux , UY ) be measurable is satisfied. Often the metric spaces chosen are one of IR, 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 = vn of cadlag functions mapping [0, oo ) into !Rn, topologized by uniform conver­ gence on compacts. While this is a complete metric space, it is not separable. Indeed, a compatible metric is d( f,

00

g) = �

1 ( 1 1\ 0 ��� l f( s ) - ( s ) l ). n 2 n

g

However if fa (t) 1 [a ,oo ) (t), then d(fa, ff3 ) = 1 /2 for all a, /3 with 0 < a < f3 < 1 , and since there are uncountably many such a , {3, the space is not separable. Fortunately, however, the condition that d(Ux, UY ) be measurable is nevertheless satisfied in this case, due to the path regularity of the functions in the function space vn. (Note that in many other contexts the space vn is endowed with the Skorohod topology, and with this topology vn is a complete metric space which is also separable; see for example Ethier-Kurtz [71] or Jacod-Shiryaev [1 10] .)

=

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 be a parameterized family of stochastic processes such that t a.s., each a E !Rn. Suppose that

f---+

E { sup IX� - X�l"' }

s
(Xf)t>O, a E JRn Xf is cadlag

< C(t)lla - b ll n+ f3

Xf

for some c, f3 > 0, C(t) > 0. Then there exists a version of Xf which is B(IR+) 181 B ( !Rn ) 181 :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 > 0, Xf

= Xf

a.s.

(The null set {Xf =1- xn can be chosen independently of t.) In this sec­ tion we will use the above corollary for parameterized processes xa which are continuous in t. In this case the process obtained from the corollary of Kolmogorov's Lemma, _X a , 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 - Bs is Gaussian with mean zero and variance t - s, we know that E{ I Bt - Bs l 4} < 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 D Corollary 1. Hypothesis A. For the remainder of this section we let X denote a semi­ martingale with the restriction that Lo < s 0.

Observe that if (n, :F, (:Ft ) t >o, P) is a probability space where (:Ft ) t >o is the completed minimal filtration of a Brownian motion B = (Bt)t >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 process. Then the jump processes dX and dA are equal, hence

FV

O< s< t since A is of finite variation on compacts. Let X be a semimartingale satisfying Hypothesis A, and let

Jt = L dX8 ; O<s< t that is, J is the process which is the sum of the jumps. Because of our hypoth­ process and thus also a semimartingale. Moreover yt = Xt - Jt esis J is an is a continuous semimartingale with Yo = 0, and we let

FV

be its (unique) decomposition, where M is a continuous local martingale and process with Mo = Ao = 0. Such a process M is then A is a continuous uniquely determined and we can write

FV

M = Xc the continuous local martingale part of X.

'

Notation. We assume given a semimartingale X satisfying Hypothesis A. If Z is any other semimartingale we write

(a E IR) .

IV General Stochastic Integration and Local Times

222

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) ®B(IR+ ) ®F measurable, cadlag version of za (cf. , 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 be a continuous local mar­ tingale with = 0, 2 < p < oo, and T a finite stopping time. Then

X X0 E{(Xr)P} < CpE{[X, X]j"2 } where Cp = {qP( p(p; l))}P/2, with ; + ! = 1 .

X and [X, X] are T T ) p(p 1 1 J Xr JP = p l{o sign(Xs) JXs JP dXs + 2 l{o JXsJP- 2d[X, X]s·

Proof

By stopping, it suffices to consider the case where bounded. By Ito's formula we have

By Doob's inequalities (Theorem

20 of Chap.

I) we have (with

! + ! = 1)

E{(Xr)P} < qPE{ J XrJ P } (p ) T = qP E{ p ; 1 1 JXs J P- 2d[X ,X ]s} < qP ( p(p; 1 ) ) E {( Xr)P - 2[X , X] r } < qP ( p(p ; 1 ) ) E{(Xr)P} l - %E{[X ,X]j"2 }2/p , with the last inequality by Holder's inequality. Since E{(X:T)P} 1 - % < divide both sides by it to obtain and raising both sides to the power p/2 gives the result.

oo ,

we

D

Actually much more is true. Indeed for any local martingale (continuous or not) it is known that there exist constants c , such that for a finite stopping time T

p Cp E{ (Xr )P } l/p < cpE{[x, xJ!f' 2 } 1/P < cpE{ (Xr )P} 1/P for 1 < p < 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 1 62 ]) . oo .

223 Theorem 74. Let X be a semimartingale satisfying Hypothesis A. There ex­ ists a version of (X c )'t such that (a, t, w) (Xc )'t(w) is B(IR) ® P measurable, and everywhere jointly continuous in (a, t). Proof Without loss of generality we may assume X - X0 E '}-{2 . If it is not, n we can stop X - Xo at rn - . The continuous local martingale part of x r ­ is then just (Xc ) Tn . Suppose - oo < a < b < oo , and let t O:t(a, b) = E{( l { b > Xs- > a } d[X, X] � ) 2 } . 7 Local Times

f---+

By Corollary 1 of Theorem

O:t (a, b) = E{(

fo

70 we have

1 Lfdu)2 } = (b - a)2 E{( b � a 1b Lfdu)2 } b < (b - a) 2 E{ b � a 1 (Lf)2du}, b

by the Cauchy-Schwarz inequality. The above implies

O:t (a, b) < (b - a) 2 u sup E{(Lf) 2 }. E(a,b) By the definition,

t { Lf < Af < I Xt - Xo l - l sign(Xs - - u)dX8 o+

and therefore

t E{(Lf) 2 } < 2E{ I Xt - Xol 2} + 2E{( l{ sign(Xs - - u)dXs)2} o+ < 4II X - Xo ll�2 + 411 sign(Xs - - u) · (X - Xo) ll�2 < BII X - Xo ll�2 < oo, and the bound is independent of u. Therefore for a constant r < oo, and independent of t. Next using Burkholder's inequal­ ity (Theorem 73) we have E{sup I (Xc ) �

s

r=

- (Xc ) �l 4 } < C4E{( o l {b> Xs- > a} d[X, X] � )2} J < c4 sup at (a, b) t < c4 r(b - a)2 .

The result now follows by applying Kolmogorov's Lemma (the corollary of Theorem 72). D

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(IR) Q9 P 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 Lta - = l"lmb-->a, b< a Ltb exts· t. Proof Since X satisfies Hypothesis A, the process Jt = Lo<s< t LlX8 is an FV semimartingale, and Y = X - J is a continuous semimartingale. We let Y = M + A be the (unique) decomposition of Y, with M0 = A0 = 0. Then X = M + A + J. Further define Sf = L l{x.->a} (Xs - a) - + L l{x._
A

A

Consider We have

and lim ( A ) f =

a-->b a< b

t}0 l{x._ >b} dAs +

where the convergence is uniform in t on [0, Tj , and T > 0. We have analogous results for ( J)f and also for Sf, because it is dominated by L o<s< t j llX, I < oo. Since we already know that ( M)t is continuous, the proof is complete. D The next three corollaries are simple consequences of Theorem 75. Re­ call that for a semimartingale X satisfying Hypothesis A, we let Jt = L o<s< t LlX., Y = X - J, and Y = M + A be the (unique) decomposition of Y, with Ao = 0. Corollary 1. Let X be a semimartingale satisfying Hypothesis A. Let X = M + A + J be a decomposition with M and A continuous and J the jump process of X, and let (Lnt>o be its local time at the level a. Then

t

1 l{x.- =a}dA, t = 2 1 l {x.= a} dA

Lf - Lf - = 2

• .

7 Local Times

225

An example of a semimartingale satisfying Hypothesis A but having a discontinuous local time is = where is standard Brownian motion = with = 0. Here for a > 0, = and + has given a more interesting (but more = 0 for a < 0. J. Walsh complicated) example.

B t Xt B l I 0 (X) L0 (B), a (X) La (B) L -a (B) L L B 0 La (X) [227] Corollary 2. Let X be a semimartingale satisfying Hypothesis A. Let A be as in Corollary 1. The local time (Lf) is continuous in t and is continuous at a = a0 if and only if 100 1{ X. =ao } ldAs I = 0. 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 aLt = El-1"m-->0 1 1t 1 { a<X 0 - 10 1 {a-E <X
-

-

8,

s -

- s_

E:

The lack of symmetry in the above formula stems from the definition of local time, where we defined sign(x) in an asymmetric way: sign(x) =

{ -1 '1 '

X > 0, X < 0.

A symmetrized result follows trivially, namely

Lta + Lta- lim -1 1t 1{ \ X. -a\ < c} d[X, X] �, a.s. 2 c---> 0 2c 0 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 t 1 Lf = E--->lim0 -2 10 1 (a-E,a+c) (Bs)ds, a.s. =

C

Local times have interesting properties when viewed as processes with "time" fixed, and the space variable "a" as the parameter. The Ray-Knight the Brownian local time Theorem giving the distribution of sampled at the hitting time of as the square of a two dimensional Bessel diffusion, is one example.

1,

(L}- a )o
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--+ in­ duces a measure on lR if is the local time of a semimartingale satisfying Hypothesis and is a positive random variable.

Lfj

L

X A U 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 f f--+ �=l fi (Lfr ' - Lfj ), i where f (x) = :L fi 1 (a; , a;+ , ( x), can be extended uniquely to a vector measure on B ( IR) with values in L0 . Proof. Recall that L0 denotes finite-valued random variables. By Theorem 68 we know that n

l

� Lf = (Xt - a) - - (Xo - a) - + Jto+ l {x._< a} dXs - � l{xs - >a } (Xs - a) - - � l{x._
D

In the proof of Theorem 76 we proved the following result, which is im­ portant enough to state as a corollary.

8 Azema's Martingale

227

U

Corollary. Let X be a semimartingale satisfying Hypothesis A, a positive Then random variable, and can be defined as an L0 -valued measure, and for f E b.B(JR.) we have

Xf J� l{x, _< a}dX8•

daXu

=

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

j

u { F(Xu ) - F(Xo) lo j(Xs_)dXs - �2 J (a)da Lu + {F(Xs) - F(Xs - ) - j(Xs_) llX8}. + O<s< U =

L

Combining Theorem 77 with the Meyer-Ito formula (Theorem 70) we ob­ tain the following relationship. Corollary. Let X be a semimartingale satisfying Hypothesis A, and let T be a finite-valued random variable. Let La be the local times of X and let f be a C1 function. Then

1: f' (a)LY,da = -1: f(a)daLY,.

Remark. The Meyer-Ito formula (Theorem 70) and the Bouleau-Yor formula can be extended in interesting ways. This was first done in [77] , which inspired subsequent work by N. Eisenbaum [62] and others (e.g. , [6]) . See [80] for a compilation of selected results in these areas and also new ones. 8 Azema's Martingale

In order to prove the regularity properties of local times in Sect. 7, we needed Hypothesis A: a semimartingale X satisfies Hypothesis if X is such that l:o<s< t l llXs l < oo a.s., each t > 0. This hypothesis was needed to prove Theorems 74 through 77 and their corollaries. We present here a counterex­ ample, known as Azema's martingale, that shows that these results do not hold for general semimartingales. In particular, Theorems 81 and 82 show that Azema ' s martingale has a local time that is zero at every level a except a = 0, and therefore there is no regular version in the space variable. Let (n, F, IF, (Bt)t>o , P) be a standard Brownian motion, Eo = 0, with the filtration IF = (Ft)t>o completed (and hence right continuous) . We define

A

228

IV General Stochastic Integration and Local Times

sign(x) =

{ -11 '

if X > 0, if X < 0.

,

Set 9� = o- sign ( Bs ) ; s < t}, and let O denote the filtration (9�) t> O completed. (It is then right continuous as a consequence of the strong Markov property of Brownian motion.) Let Mt = E{Btl9t}, t > 0. Then for s < t,

{

and M is a
Definition. The process Mt

=

E{Btl9t} is called Azema's martingale.

Fundamentally related to Azema's martingale is the process which gives the last exit from zero before time t. We define

9t = sup{ s < t Bs = 0}. :

To study the process g we need the reflection principle for Brownian motion, which we proved in Chap. I (Theorem 33 and its corollary) and which we recall here for convenience. Theorem 78 (Reflection Principle for Brownian Motion) . standard Brownian motion, B0 = 0, and c > 0. Then P(sup B8 > s
Let B be

c) = 2P( Bt > c) .

An elementary conditioning argument provides a useful corollary. Corollary. Let B be standard Brownian motion, B0 = 0, and 0 < s < Then P( sup Bu > 0, Bs < 0) = 2P( Bt > 0, Bs < 0). s
The process 9t is
<

s)

=

t.

; arcsin -Jt,

Proof. For s < t we have almost surely uE s
uE s
from which it follows that 9t is 9t adapted. To calculate the distribution of 9t we observe for s < t (using the symmetry of B),

8

Azema's Martingale

229

P(gt > s) = 2P(gt > s, Bs < 0) Bu > 0, B8 < 0) = 2P( s sup 0, Bs < 0), by the corollary of Theorem 78. Since (B , Bt) are jointly Gaussian, there exist X, Y that are independent N(O, 1) random variables with 8

and

Bt = vs X + � Y.

Then {Bs < 0} = {X < 0}, and {Bt > 0} = {Y > - .#-s X}. To calculate P( {Y > - #s X, X < 0} ) , use polar coordinates:

1 7r-arcsin 1 oo

J"f 1 - r2 12 rdrd() P (Bt > 0, Bs < 0) = e 2w ,. /2 w - arcsin fs) = ...!_ ( 2w 2 Vt � - ...!.._ arcsin fs_ 4 2w Vt Therefore 4P ( Bt > 0, Bs < 0) = 1 - ; arcsin v1 and P(gt < s) = o

=

; arcsin v't.

0

Theorem 80.

Azema 's martingale M is given by Mt sign(Bt) �Jt - gt > =

t > 0. Proof. By definition,

However E{IBtll9t} = E{ I Btll9ti sign(Bs), s < t }. But the process sign(Bs) is independent of !Btl for < 9t, and sign(Bs) = sign(Bt) for 9t < s < t. Hence sign( B ) is constant after 9t. It follows that 8

8

Therefore if we can show E{IBtl l9t = s} = ��' the proof will be complete. Given 0 < < t we define 8

T = t !\ inf{ u > s Bu = 0}. :

Note that T is a bounded stopping time. Then

230

IV General Stochastic Integration and Local Times

E{IBt l i 9t

< s} = 2E{Bt i 9t < s = 2E{Bt ; T = t

and Bs > 0} and Bs > O}

= 2E{Br ; Bs > 0}

= 2E{Bs ; Bs > 0},

=

since Br 0 on {T of t}, and the last step uses that B is a martingale, and Therefore s < T. The last term above equals E{IBs l } =

{f.;s.

E{IBt I I 9t

= s} = fs E{P(I Bt9lt; gt<- :::;) s} = fi�, JJ.. ds

since fs

S

P(gt < s) = ,. Js1(t - s) , by Theorem 79.

0

Let H? = a{M8; s < t }, the minimal filtration of Azema's martingale M, and let lHI = be the completed filtration.

('Ht)t>O

Corollary. The two filtrations lHI and
By definition Mt E 9t, each t > 0. Hence 1it Theorem 80, sign(Mt) sign(Bt), so that 9t C 1it.

Proof.

c

9t ·

However, by

= 0 Next we turn our attention to the local times of Azema's martingale. For a semimartingale (Xt)t >o, let (Lf(X)) t > O = La (X) denote its local time at the level a. Theorem 81.

zero. Proof.

The local time at zero of Azema 's martingale is not identically

=

By the Meyer-Ito formula (Theorem 70) for f(x) lxl we have t sign(Ms_ )dMs + L� (M) + L { I Ms i - I Ms - l - sign(Ms - )6.Ms} · 0

=1 O<s
yt

The local times at all levels except zero of Azema 's martingale are 0. That is, Lf (M) = 0, t > 0, if a of 0.

Theorem 82.

8

Azema's Martingale

231

By Theorem 69 we know that the measure dL�(w) on JR. + 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 0. Since L0 = 0, we have Lf 0 0 for a of 0.

Proof

=

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, L0 (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.

The local time L0 (B) of Brownian motion is
Theorem 84.

Theorem measurable functionals of the zero set of Brownian motion. Such a result is classical: see, e.g. , Kingman [126, page 730] . Theorem 85. The local times at zero of Brownian motion and Azema 's mar­

tingale are the same. That is, L0 (B) = L0 (M) . Proof. As we saw in the proof of Theorem 81, IMtl - L�(M) is a martingale. Since the process L0 (M) is non-decreasing, continuous and adapted, it is natural. If A is another natural process such that I M I - A is a martingale, then L0 (M) = A by (for example) Theorem 30 of Chap. III. Thus, it suffices to show that I MI - L0 (B) is a martingale for the filtration G. By Tanaka's formula, I Bt l - L�(B) = Nt , where N is an IF martingale. Then

and by Theorem 84

E{I Btll9t} - L�(B) is a martingale. Since

E{ I Bt l l9t} = E{sign (Bt)Btl9t } = sign (Bt)E{Btl9t } = sign (Bt)Mt = I Mtl, we have I Mt l - L�(B) is a martingale, and the theorem is proved.

0

IV General Stochastic Integration and Local Times

232

Corollary 1. The process

Proof

I MI - L0 (B) is a G martingale.

This corollary is proved at the end of the proof of Theorem 85.

0

l{ B, >o} .;Ivt - gt - �L�(B) is a o} M; = l{M < o } M; = 0, it follows from Theorem 68 that M/ - �L� (-!\-1) is a martingale. However Mt+ = l { B, >O} �vt - gt by Theorem 80, and L�(M) = L�(B) by Theorem 85, and the theorem follows. Corollary 2. The process

.

_

0

The next theorem shows that Azema's martingale is quadratic pure jump. Theorem 86.

Proof.

For Azema 's martingale M, [M, M]c

By Corollary 2 of Theorem 70,

[M, M]f =

=

0,

and [M, M] t = � gt.

1: Lfda.

By Theorem 82, [M, M] c = 0. We conclude that [M, M]t and it follows from Theorem 80 that [M, M]t = ; gt.

=

L o<s
0

Theorem 86 allows us to compute the compensation of [M, MJ; that is, the unique predictable, increasing process A such that [M, M ] - A is a martingale. This process A is denoted (M, M) in the literature, and it was defined in Sect. 4 of Chap. III.

Let M be Azema 's martingale. Then [M, M]t - � t is a mar­ tingale. That is, (M, M)t = � t. Proof Recall that Ml - [M, M] t is a martingale ( cf. , Theorem 27 of Chap. II) .

Theorem 87.

By Theorems 80 and 86,

Mt2 - [M, M]t = ; {(t - gt) - gt } 7r = 2 t - 7rgt . Since the above is a martingale, it remains only to observe that ; gt - � t is a 0 martingale.

Note that if Xt = sign (Bt ) V2y't - gt . then X is a quadratic pure jump martingale with (X, X)t = t. Recall that if N is a Poisson process with intensity one, then Nt - t is a martingale. Since [N, N]t = Nt, we have [N, N]t - t is a martingale; thus (N, N) t = t. Another example of a martin­ gale Y with (Y, Y)t = t is Brownian motion. Since [B , B]t = t is continuous,

[B, B]t = (B, B)t = t.

9 Sigma Martingales

233

9 Sigma Martingales

We saw with Emery's example (which precedes Theorem 34) a stochastic integral with respect to an 7t2 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 J.k on a space such that there exists a sequence of measurable sets Ai E l3 such that Ui Ai = and M(Ai ) < oo, 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 JR.d-valued semimartingale X is called a sigma martingale if there exists an JR.d-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 1t1 .

(S, B) S

·

·

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 M0 = 0. 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 7t1 by Theorem 51, so let Tn be a sequence of stopping times tending to oo a.s. such that M Tn E 7t1 for each n. Set To = 0 and let Nn = lcrn- t ,Tn] · M Tn . Let O:n be a strictly positive sequence of real numbers such that Ln n:n ii N n l l w < oo. Define N = Ln o:n Nn and one can check that N is an 7t1 martingale. Set = l { H O } + H L n an 1l (Tn -t ,Tn ] . Then it is simple to check that X = N and that is strictly positive. 0

J

J

·

=

J

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 xrn is a sigma martingale for each n. Since x rn is a sigma martingale, there exists a martingale M n in 7t 1 such that x rn = Hn · Mn , for each where Hn is positive predictable. Choose ¢n positive, predictable so that ll¢n Hn · MTn ll w < T n . Set To = 0 and ¢0 = ¢ 1 1 {0 } · Then ¢ = ¢0 + Ln > l n l(Tn -t,Tn ] is strictly positive and predictable. Moreover ¢ · X is an 7t1 martingale, whence X = � · ¢ X, and 0 the corollary is established.

n,

·

234

IV General Stochastic Integration and Local Times

Corollary 2. A local martingale is a sigma martingale. 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

Proof

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 H 1 = H1 {H>O} + 1 and H2 = -H1 {H are both strictly positive for i = 1, 2 we have by Theorem 88 that Hi ¢> · M is a sigma martingale for each i = 1 , 2. Sigma martingales form 0 a vector space, and we are done. We also would like to have sufficient conditions for a sigma martingale to be a local martingale. Theorem 90. A sigma martingale which is also a special semimartingale is

a local martingale. Proof Any sigma martingale is a semimartingale by definition; here we also assume it is special. Thus it has a canonical decomposition X = M + A where M is a local martingale and A is a process of finite variation on compacts which is also predictable. We want to show A 0. We assume A0 = 0. Choose H predictable, H > 0 everywhere, such that H · X is a martingale, which we can easily do using Theorem 88. Then (H 1\ 1) · X = ( Hfi 1 ) . (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 = 0. If we replace H with JH where J is predictable, IJI < 1, and is such that J� J8dA8 = J� l dA8 I, then we conclude that J� H8 ldA8 I = 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

X has continuous paths, then it is a local Corollary 2. If X is a sigma martingale and if either x; SUPs
=

=

Bibliographic Notes

The extension by continuity of stochastic integration from processes in lL to predictable processes is presented here for essentially the first time. How­ ever the procedure was indicated earlier in Protter [201, 202] . Other ap­ proaches which are closely related are given by Jacod [ 103] and Chou-Meyer­ Stricker [31] (see an exposition in Dellacherie-Meyer [46, page 381] ) . The "1i" in the space 1i2 of semimartingales comes from Hardy spaces, and this bears explaining. When the semimartingale E 1i2 is actually a martingale, the space of JiP martingales has a theory analogous to that of Hardy spaces. While this is not the case for semimartingales, the JiP norms for semimartingales are, in a certain sense, a natural generalization of the JiP norms for martingales, and so the name has been preserved. The JiP norm was first introduced by Emery [64] , though it was implicit in Protter [197]. Most of the treatment of stochastic integration in Sect. 2 of this chapter is new , though essentially all of the results have been proved elsewhere with dif­ ferent methods. The author benefited greatly throughout by discussions with S . Janson. Theorem 1 (that 1i2 is a Banach space) is due to Emery [66] . The equivalence of the two pseudonorms for 1i2 (the corollary of Theorem 24) is originally due to Yor [241] , while the fact that 1i2 (Q) c 1i2(P) if �� is bounded is originally due to Lenglart [144] . Theorem 25 is originally due to Jacod [103, page 228] . The concept of a predictable o--algebra and its impor­ tance to stochastic integration is due to Meyer [166] . The local behavior of the stochastic integral was first investigated by McShane for his integral [154] , and then independently by Meyer [171], who established Theorem 26 and its corollary for the semimartingale integral. The first Dominated Convergence Theorem for the semimartingale integral appearing in print seems to be in Jacod [103] . The theory of martingale representation dates back to Ito's work on mul­ tiple stochastic integrals [96], and the theory for M2 presented here is largely due to Kunita-Watanabe [134] . A more powerful theory (for M 1 ) is presented in Dellacherie-Meyer [4 6] . Absolutely continuous spaces are defined here for the first time. The author benefited from discussions with Yan Zeng on this subject. That the dual of 1i 1 is BMO for martingales is due to the combined

X

236

Exercises for Chapter IV

efforts of many researchers, culminating in Meyer's paper [170] ; we follow the treatment in [87] . The Jacod-Yor Theorem is from [1 1 1] , 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 Doleans­ 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 [119] 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 J . See also Millar [181] . 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 [243] 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 [38] . See [1 10] for a more comprehensive treatment . Exercises for Chapter IV

Exercise 1. Let X be a semimartingale. Show that X is special if and only [X, X ] t is locally integrable; that is, there if the increasing process Ct exists a sequence of stopping times Tn increasing to infinity a.s. such that E{ [X, X] rn } < oo for each time Tn*Exercise 2. Let X be a special semimartingale with canonical decomposi­ tion Xt = X0 + Mt + At . Show that the following two inequalities hold: =

E{[A, A] oo }

< E{ [X, X] oo } and E{ [M, M] oo } < 4E{ [X, X]00}.

Exercises for Chapter IV

237

Xn n.

Exercise 3. Let (D, :F, P) be a complete probability space. Let be a sequence of random variables such that < oo ) = 1, all 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 �� bounded, and every random variable is in L1 (dQ). Exercise 4. Let be a semimartingale. Show that there is a probability measure Q, equivalent to P, such that under Q, has a decomposition + A, where is a martingale with < oo for each t, 0 < t < oo, and A is a predictable process of finite variation on compacts with fdA s f ) < oo for each t, 0 < t < oo. (Hint: Use Exercises 2, and 3.) *Exercise 5. Let be a sequence of semimartingales. Show that there exists one probability measure Q, equivalent to P, such that under Q, each has a decomposition is a martingale with = where < oo for each t, 0 < t < oo and each and is a predictable fdA� I ) < oo for each t, process of finite variation on compacts with 0 < t < oo and each 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.) = 0 for Exercise 7. Let be a Levy process with < oo and all t > 0. Show that is a martingale, and that ze is either 0 or Brownian motion, where ze is defined in Exercise 6. Conclude that any martingale Levy process that has no Brownian component is purely discontinuous. Exercise 8. Let N be a standard Poisson process with arrival intensity >.. Show that the martingale Nt - >.t 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 be a square integrable martingale. Show that = e 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 is an arbitrary square integrable martingale, then its orthogonal decomposition with respect to this subspace Give a description of the orthocomplement of S. is M = Exercise 1 1 (example of orthogonal projection) . Let T be a totally be a square integrable martingale. Let inaccessible stopping time and let and let L be the martingale of minus its compensator. We denote L by II Show that IIM and II 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 II represents the orthogonal projection of onto the orthocomplement of S.

P(f Xn f

Xn X M

M EQ(J;

xn

xn n n EQ ([M , M ]t)

xn Mn + An ,

1,

Mn n n, A EQ(J;

n.

E{ I Ztl}

ZZ

Z

X=

X EQ([M, M]t)

E{Zt}

[M, M]f

M [Me, Me]t = [M M]t, ,

M

MT + (M - MT ).

Ut = .6.Mr 1 {t > T}

M

M.

U M M -

M

M

238

Exercises for Chapter IV

X

* Exercise 12. Let T be an arbitrary stopping time, T > 0, and let be = Show that is a uniformly integrable a random variable in = 0. martingale if and only if

£1.

> M } X1{ t T E{X I Fr- }

* Exercise 13. Let T be a stopping time, T > 0, and let integrable martingale. Show that the decomposition

M be a square

is an orthogonal decomposition. Show that one of the stable subspaces in is such that question is the space of all square integrable martingales measurable. Use Exercise 12.)

M

Mr

(Hint: Fr*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 l X.x2t for every >. -::/:- 0. Note that by the uniqueness in distribution of the solution, the processes (Xt)t>O and (lX.x2t)t>o have the same distribution.

X M M 1 E{Moo} ci M I rt1 M.

*Exercise 16. Let be a martingale solution of the structure equation of Exercise 15. Show that xc = 0, where xc is defined in Exercise 6. for some p > 1. Show be a local martingale in Exercise 17. Let that is a uniformly integrable martingale. Show further that if p = 1 then < where the constant is universal. That is, does not depend on Exercise 18. Suppose that

c

1{P

c

M is a bounded martingale. Show that

This can improved to replace J5 with 2.

BMO H H M.

Exercise 19. Show that there are martingales with finite norm which are not bounded. Let be a bounded martingale and be a pre­ < 1, and let N = · Then it is easy to see dictable process such that that N is in and to construct such an N that is not bounded.) Exercise 20. Prove this version of the Fundamental Theorem of Local Mar­ tingales. Let be a uniformly integrable martingale. Show that there exist N is of integrable variation, and = + V. and such that N is in If is a local martingale this holds locally.

U M

(Hint: M H I I BMO, M BMO, U

M U

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, is a local martingale. Give then E bP so that Mt- = an example of a local martingale M and a stopping time T such that MT- is not a local martingale. Exercise 23. Let be a continuous semimartingale. Show that = Use local times. ) *Exercise 24. Let be standard Brownian motion and let o: > 0. Show that Use local times. ) f01 < a.s. if and only if o: < 1. be a semimartingale. Show that Exercise 25. Let = denotes the local time of at time t and where 0) and level 0. Exercise 26. Let A be adapted, continuous, and of finite variation on com­ pacts, and let be standard Brownian motion. Let >. denote Lebesgue mea­ = A = 0 a.s. sure on the line. Show that >.(s : Exercise 27. Let be a continuous semimartingale. Show that E denotes the smallest right continuous completed fil­ each > 0, where tration generated by the process Exercise 28. Let be a semimartingale. Show that for each fixed (w, the section �----> has compact support. Exercise 29 (Skorohod's Lemma) . This result will be useful for the next such that two exercises. Let be a real valued continuous function on [0, > 0. Show there exists a pair of functions y on such that =y y is positive, and is increasing, continuous, = 0, and the induced by is carried by the set {s : y(s) = 0}. Show further measure sup that the function is given by s V 0) . 0. Let Exercise 30. Let M be a continuous local martingale with M M) . Show that is a continuous local martingale with = = sup = 0, and that * Exercise 31. Let be a continuous semimartingale, and suppose satisfies . Show that the equation = sign where = Use Tanaka's formula and Exercise 29.) Exercise 32. Suppose that and Y are continuous semimartingales such = 0. Show that that

J; l [o,r)(s)dM8

l [o,r) (s)

X [X, X]. (Hint: B I Bt l - "' dt oo X x + = max(X,

[ l X I, X]

(Hint:

L� (X)

B

t

x

Bs s)

XX .rJ I X Lf(X)

(I Xtl)t> O ·

L� (X) �L� (X + ) X L� .rJ X I , t)

oo) a) [ O , oo) ( x(O) + a , a (O) a x , da8 a a a(t) = sO }dL�(X) + lo l{Xs >o}dL�(Y) + lo l{Xs=Ys=O} dL�(Y+ - X + ). x

s

-

V

A

Exercises for Chapter IV

240

X and Y be continuous semi­ L�(X Y) + L�(X 1\ Y) L�(X) + L�(Y). *Exercise 35 (Emery-Perkins Theorem) . Let B be standard Brownian L� Let X B + cL, c E JR. fixed. Assume that motion and set Lt (B). there exists a set D JR.+ D such that (i) D is predictable for :Fx ; (ii) each t > 0; and (iii) P(w : (Tt(w),w) E D) 0 each t 1 E D) P(w : ,w) ( t > 0, where Tt inf(s > 0 Ls > t). (a) Show that Bt J� 1D(s)dX5 • Also show that Bt E :Fx , each t > 0, and conclude that :FE :Fx :FB + cL , all c R (b) Let Exercise 34 (Ouknine's Formula) . Let martingales. Show that v

=

= c

=

x

=

=

=

:

=

D

=

{ (t,w) :

E

=

=

lim

� L....t 1{ x

- k (w} -X

- ( k - t J (w}> O}

=

I_2

}

·

k=l Use the fact that s (BsA T, Lsi\ T, ) and s (-Br, _s,L r, -L r, _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 I B I + cL. Show that :FY :FI B I , for all c. n-->oo

t-2

�---->

t-2

�---->

c

=

*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 :FY , with

C-1 -

'

yt f3t + Jot Y1s ds for an :FY -Brownian motion (3. Use this to show that (Note: One can show that :FY :FIBI for all c 1.) =

=

=!=

:FY

-::f. :FIB I for

c 1. =

*Exercise 38 (Barlow's Theorem) . Let X solve the equation

B

where is a standard Brownian motion. The above equation is said to have and Y are two solutions, and if a unique weak solution if whenever �----> = /l Y · The is defined by !lx (A) = E A}, then above equation is said to have a unique strong solution if whenever both and are two solutions on the probability space on which the Brownian = 1. Assume that motion is defined, then �----> = t �----> are the equation has a unique weak solution with the property that if any two solutions defined on the same probability space, then for all > 0, = 0. Show that the equation has a unique strong solution.

11x X Y B

L�(X - Y)

X P{w : t Xt(w)

P{w : t Xt(w)

11x

yt(w)}

X, tY

Exercises for Chapter IV

241

Exercise 39. Consider the situation of Exercise 38, and suppose that and b are bounded Borel. Further, suppose that < for all where [0, oo ) [0, oo ) is increasing with J;+ p(�) + oo for every E > 0. Show that if Y are two solutions, then Y) 0.

x, y,

p:

(u (x) - u(y))2 p( l x - Y l) du = L�(X - =

-+

u

X, *Exercise 40. Let B be a standard Brownian motion and consider the equation t Xt = 1 sign(Xs)dB5• 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, (Hint: show that if X is a solution, then X is not adapted to the filtration Show first that Bt = J; 1{1 Xsi #O} d 1 Xsl · ) Exercise 41. Let X be a sigma martingale and suppose X is bounded below. Show that X is in fact a local martingale. :FE .

*Exercise 42. Show that any Levy process which is a sigma martingale is actually a martingale (and not just a local martingale) .

be i.i.d. random variables with P(U1 = 1) = (U >l ) n n P(Ul = -1) = 1/2, and let X = En> l 2-n un 1{t> qn} where (qn ) n> l is an enumeration of the rationals in (0, 1). Let Ht = En > l � 1 {t > qn } and show that X E 1{2 and H E L(X), but also that X is of finite variation and Y = H · X has infinite variation. Thus the s ace of finite variation processes Exercise 43. Let

is not closed under stochastic integration!

p

xn P) P, 1{2. xn Lx B x. A = {w : dL�(w) is singular as a measure with respect to ds} where of course ds denotes Lebesgue measure on JR. + . Show that P(A) = 1. Conclude that almost surely the paths s 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 Xtway.= (Bt, NLJ. Let IF be the minimal filtration of X, completed in the usual (a) Show that X is a strong Markov process.

*Exercise 44. Let be a sequence of semimartingales on a filtered complete probability space (D, :F, IF, satisfying the usual hypotheses. Show there ex­ ists one probability Q, which is equivalent to with a bounded density such is a semimartingale in that under Q each Exercise 45. Let be a standard Brownian motion and let be its local time at the level Let

f---+

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 = L t/\ T ·

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 4 7. 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. 1.)

v Stochastic Differential Equations

1 Introduction

A diffusion can be thought of as a strong Markov process ( in JR.n ) 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 1 of the form

where B is a Brownian motion in JR.n , 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 (.Ft)t >O is the underlying filtration for the Brownian motion B, then for small E > 0

E{Xf+c: - Xfi.Ft} = bi (Xt)E + o (c) E{ (Xf+c: Ebi (Xt))(Xl+c: - xt - c&i (Xt))I.Ft} = (uu ') i1 (Xt)E + 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, Bt = J; �5ds, an equation

x: -

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 X0 = 0 and canonical decomposition X = N + A, then We now use an equivalent norm. To avoid confusion, we write H2 instead of 1{2 . Moreover we will define HP, 1 < p < oo. We begin, however, with a different norm on the space lDl (i.e. , the space of adapted cadla,g processes) . For a process H E lDl we define IIHII sv

=

IIH * II LP ·

Occasionally if H is in lL (adapted and caglad) we write IIHII-sv as well, where the meaning is clear.

2 The HP Norms for Semim ti gal ar

A

n

es

245

If is a semimartingale with paths of finite variation, a natural definition where of a norm would be denotes the total variation measure on JR. + induced by s �----> Since semimartingales do not in general have such nice paths, however, such a norm is not appropriate. 0, Throughout this chapter, we will let denote a semimartingale with 0). By the Bichteler­ a.s. Let be an arbitrary semimartingale (with Dellacherie Theorem (Theorem 43 of Chap. III) we know there exists at least one decomposition with a local martingale and an adapted, cadlag process, with paths of finite variation (also 0 a.s. ) . For 1 < p < oo we set

Z

I A I P = I j000 l dAsii i Lv, As (w). ldAs(w)l Z Z = 0 Z0 = Z = N +A, N A N0 = A0 =

Z be a semimartingale. For 1 < p < oo define I Z I =Hv = inf jp( N, A) where the infimum is taken over all possible decompositions Z = N +A where martingale, A E with paths of finite variation on compacts, NandisAoa local = No = 0. Definition. Let

Z=N+A

IDl

The corollary of Theorem 1 below shows that this norm generalizes the 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 cf. , e.g., Dellacherie-Meyer

HP

(

[46]).

(Zo = 0) . Then I [Z, Z]�2 1 Lv < I ZI I HP , Proof Let Z = M + A, Mo = Ao = 0, be a decomposition of Z. Then [Z, Z] �2 < [M, M] �2 + [A, A] ;b 2 = [M,M] �z + (ls ).6.As) 2 ) l/ 2 s Let Z be a semimartingale (1 < p < oo ) .

Theorem 1.

A is a quadratic pure jump semi­ LP I [Z,Z]�2 1 Lv < jp(M,A) and the resultD Corollary. If Z is a local martingale ( Zo = 0), then I Z I Hv = I [ Z, Z];b 2 J I £ v .

where the equality above holds because norms yields martingale. Taking

��-

246 V Stochastic Differential Equations Proof. Since Z is a local martingale, we have that Z Z+0 is a decomposition of Z. Therefore I Z I HP < jp(Z, 0) I [Z, Z]�,:?lb· By Theorem 1 we have I [Z, Z] ZxS2 I L v < I Z I w , 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 < oo there exists a constant such that for any = 0, semimartingale - < Proof A semimartingale Z is in lDl, so makes sense. Let = +A = be a decomposition with = 0. Then

Cp

I ZI I sp cp i ZI I Hv . M Z I ZI I sv Mo A0 00 00 I Z I �v E{(Z�)P} < E{(M� + 1 l dAs i )P} < epE{(M�)P + ifo 1 dAsi ) P}, using (a + b)P < 2P - 1 ( aP + bP ) . In the case 2 we have by Doob's maximal quadratic inequality that E{(M�) 2 } < 4E{ M�J 4E{[M, M] 00 }. Z, Zo

=

p

=

=

For general p ,

1 < p < oo, we need Burkholder's inequalities, which state

Cp

for a universal constant which depends only on p and not on the local martingale For continuous local martingales and p > 2 we proved this using Ito's formula in Chap. IV ( Theorem 73 ). For general local martingales page 28 7 . and for all finite p > 1 see, for example, Dellacherie-Meyer Continuing, letting the constant vary from line to line we have

M.

Cp 00 < I Z I �v CpE{(M� )P + ( fo l dAs i )P} 00 < CpE{[M, MJ:2 + ( 1 l dAsi )P} < Cp[)p(M,A)JP,

[46,

and taking p-th roots yields the result.

HP

]

0

Corollary. On the space of semimartingales, the - norm is stronger than the norm, 1 < p < oo.

SP

Theorem 3 (Emery's Inequality ) .

Let Z be a semimartingale, and ! + ! = � (1 < p < oo, 1 < q < oo) . Then

00 1 1 HsdZsl l w < I HI I sv i Z I Hq ·

H E lL,

2 The HP Norms for Semimartingales

247

>O· Recall that we always assume Z0 = 0 H Z MdenoteA be(J�aH8dZ8)t decomposition of Z with Mo Ao 0 a.s. Then Z + H · M + H · A is a decomposition of H · Z. Hence II H · Z l w < jr (H · M , H · A) . Next recall that [H · M, H M] = J H;d[M, M]8, by Theorem 29 of Chap. II. Therefore oo Jr (H · M,H · A) = 1 (100 H;d[M,M]s) 1 12 + 1 I Hs l ldAsl l u < II H�([M , M] �2 + 1 00 l dAs l ) l u < I H� I LP I ([M, M] � 2 + 1 00 l dAsi) I Lq = I H I sPJq (M, A), ·

Proof.

Let a.s., and let

=

=

=

·

where the last inequality above follows from Holder's inequality. The foregoing implies that

I H · Z l w < I H I sPjq (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 lJ]l and a stopping time T, recall that

+ +

xT = Xtl[o,T) Xrl [T, oo) 1 Xtl[o,T) Xr _ l [T,oo ) ·

xT- =

A property holding locally was defined in Chap. I, and a property holding prelocally was defined in Chap. IV. Recall that a property 1r is said to hold has property for each n, where rn locally for a process X if is a sequence of stopping times tending to oo a.s. If the process X is zero at zero (i.e., Xo = 0 a.s. ) then the property 1r is said to hold prelocally if xr _ has property 7r for each n.

xrn 1{Tn>O }

7r

n

SP rn HP ) x 1 {Tn> O}

( resp. Definition. A process X is locally in if there exist stopping is in ( resp. times Tn n l increasing to 00 a.s. such that for each n, < p < oo If Xo 0 then X is said to be prelocally in (resp. if xr n _ is in ( resp. for each n.

( ) 1>

SP HP) = SP SP HP) HP) 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 ( Z0 0). Then Z is locally in HP, 1 < < .

=

p

00

.

pre

248

V Stochastic Differential Equations

25 43 Z = M +A, M0 = A0 =

Proof.. By the

Fundamental Theorem of Local Martingales (Theorem of Chap. III) and the Bichteler-Dellacherie Theorem (Theorem of Chap. III) we know that for given E: > 0, has a decomposition 0 a.s., such that the jumps of the local martingale M are bounded by E:. Define inductively

Z

To = 0, t Tk+ l = inf{t > Tk : [M,MJ: 12 + lo i dAs l > k + 1 } . (Tk ) k> l are stopping times increasing to oo a.s. Moreover zTk - = (MTk ) + (Ark - - �Mrk l [rk ,oo) ) = N + C

The sequence

zrk - . Also, since [M, M] rk = [M,M] rk - + (�Mrk )2, 00 Joo (N, C) = I [N,N] � 2 + 1 l dCsl l u"' Tk = I ([M, M]rk - + (�Mrk ) 2) 112 + J{o l dCsii i Loo < ll (k2 + c2) 1/2 + (k + c)I I Loo < Therefore zrk- E H 00 and hence it is in HP as well, 1 < p < Definition. Let Z be a semimartingale in H00 and let a > 0. A finite sequence < Tk is said to a-slice Z if Z = zrk­ of stopping times 0 = To < T1 < and I ( Z - zT; ) T; + t - u H oo < a, 0 < i < k - 1 . If such a sequence of stopping times exists, we say Z is a-sliceable, and we write Z E S(a). Theorem 5. Let Z be a semimartingale with Z0 = 0 a.s. (i) For a > 0, if Z S(a) then for every stopping time T, zr E S(a) and zT- E S(2a) . (ii) For every a > 0, there exists an arbitrarily large stopping time T such that zT- E S(a) . Proof. Since zr- = MT +(AT- -�Mrl [T, oo) ), and since I ZT I Hoo < I Z I -H 00 always, one concludes I ZT - I Hoo < 2 I ZI I-Hoo, so that (i) follows� Next consider (ii) . If semimartingales Z and Y are 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. is a decomposition of we conclude

oo

.

00 .

·· ·

E

D

3 Existence and Uniqueness of Solutions

J;k

249

J�

ldAs l > a/8 or ldAs l > k}. For A, let To = 0, Tk+ l = inf{t > Tk : E S(a/8) for each k, and the stopping times (Tk) increase to oo Then a.s. > For M, let Ro = 0, Rk+l = inf{ t > Rk E each k, and moreover or t > k}. Then

ATk -

MRk - H00 ,

(32 [M, M]

:

[M,M]t - [M,M] Rk

Hence

I (M-MRk ) Rk+ t - I Hoo < I ([M, M] Rk + t - [M, M] Rk ) 112 + � �MRk +lii i L00 = I ((�MRk + 1 )2 + [M, M] Rk+ l - - [M,M] Rk ) 1 12 + � � MRk+tii i L00 < 1 ((32 + (32 ) 1 /2 + fJ I Loo = (1 + v'2)(3. Thus for each k, M R k - E S( (1 + v0,)(3), and since (3 a/24, the result follows.

D

3 Existence and Uniqueness of Solutions

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 45 1 . Second, we consider only Lipschitz-type hypotheses and thus obtain strong solutions. There is a vast literature on weak solutions cf., 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 Xt 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 is in L if it has caglad paths and is adapted.

]

(

H

Theorem 6. Let Z lR be such that

be a semimartingale with Z0

(i) for fixed x, (t, w)

f---+

j(t, w, x) is in L; and

=

0

and let f : JR+ x f2 x lR

---->

250 V Stochastic Differential Equations (ii) for each (t, w), lf(t, w, x)-f(t, w, y)l < K(w)lx-yl for some finite random variable K. Let X0 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 lR is Lipschitz if there exists a (finite) constant k such that (i) lf(t, x) - f(t, y)l < klx - yi, each t E JR+ , and (ii) t f(t, x) is right continuous with left limits, each x E JRn . ---+

:

1---t

f is said to be autonomous if f(t, x)

=

f(x), all t > 0.

Definition. A function f JR + X n X ]Rn lR is random Lipschitz if f satisfies conditions (i) and (ii) of Theorem 6. Let IIlln denote the space of processes X = (X . . . , x n ) where each X i E IIll :

-

1,

(l < i < n).

Definition. An operator F from IJ]ln into 1Ill 1 = lJ]l is said to be process Lipschitz if for any X, Y in IJ]ln the following two conditions are satisfied: (i) for any stopping time T, xr- = yr- implies F(Xf (ii) there exists an adapted process K E L such that

=

F(Yf- , and

Definition. An operator F mapping IJ]ln to = lJ]l is functional Lipschitz if for any X, Y in IJ]ln the following two conditions are satisfied:

1Ill1

(i) for any stopping time T, xr- = yr- implies F(X)T- = F(Yf- , and (ii) there exists an increasing (finite) process K = (Kt)t>O such that IF(X)t F(Y)t l < Kt iiX - Y ll; a.s., each t > 0.

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 = (At)t>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) = j (t, w; Xs , s < t), where f is defined on [0, t] X n X D[O, t] for each t > 0; here D[O, t] denotes the space of cadlag functions defined on [0, t]. An­ other example is a generalization of the coefficients introduced by Ito and Nisio [101], namely

F(X)t =

t

lo g(u, w, Xu )JL(w, du)

for a random signed measure J.l and a bounded Lipschitz function g with constant C(w) . In this case, the Lipschitz process for F is given by Kt (w) = C(w)II JL (w)t ll , where II JL (w)tll denotes the total mass of the measure p(w, du) on [0, 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 < oo, let J E let F be functional Lipschitz with F(O) = 0, and suppose supt I Kt(w) l < k a.s. Let Z be a semimartingale in Then the equation H= such that IIZ I IHoo <

2c�k .

Xt = Jt + has a solution in

SP , t

lo F(X) s dZs _

SP. It is unique, and moreover

< 2IIJ IIs-p . II X IIsp Proof. Define A SP SP by A(X)t = Jt + f� F(X)8dZ8• Then by Theo­ rems 2 and 3 the operator is 1/2 Lipschitz, and the fixed point theorem gives :

----+

existence and uniqueness. Indeed

J

I I X II sp < I I JII sp + II F(X ) s _ dZs ll sp < IIJII -sp + cp ii F(X)II -sp ii ZI IH-00 < I I J II sp + 2k1 IIF(X) II sp .

252

V Stochastic Differential Equations , we have IIXII sp < I I J IIsp + � II XIIsP , \IF( X) I I s p = II F(X ) - F(O ) IIsP -

Since which yields the estimate.

0

Lemma 2. Let 1 < p < oo, let J E SP, let F be functional Lipschitz with F(O) = 0, and suppose supt I Kt (w)l < k < oo a.s. Let Z be a semimartingale such that Z E S( �k ) . Then the equation

2c

t

Xt = Jt + lo F(X) s-dZs has a solution in SP. It is unique, and moreover I I X I Isp C(k, Z) is a constant depending only on k and Z.

< C (k , Z)IIJ IIs P , where

Proof Let z = II Z IIH oo and j = IIJ IIsP . Let 0 = To , Tb · · · , 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 0 since J 0 - = zo - = 0 for all t, and its SP norm is 0. Assume that equation ( i) has a unique solution X i , and let x i = I I X i llsp . Stopping next at Ti instead of Ti -, let yi denote the unique solution of Yi = JT, + J F(Y i )8_dz'[' , and set y i = II Yi llsp . Since yi = X i + {�Jr, + F(X i)r, - �ZrJl[r, ,oo) ' we conclude that II Y i lls-p < IIX i ll -sp + 211 JI Is-p + II F(Xi )ll sp i iZ II Hoo < x i + 2j + kx i z = x i (1 + kz) + 2j; =

-

-

hence We set for U E IJ]l, Di U = (U - UT' f' + ' - . Since each solution X of equation (i + 1) satisfies xr, = yi on [0, Ti+t), we can change the unknown by U = X - (Yi f' +' - , to get the equations = Di J + F(Yi + U) s - dDi Z8• However since F(Yi + 0) need not be 0 we define Gi (·) = F(Yi + · ) - F(Yi ) , and thus the above equation can be equivalently expressed as

U

J

We can now apply Lemma 1 to this equation to find that it has a unique solution in SP, and its norm ui is majorized by

j

ui < 2 II DJ + F(Yi ) s - dDi Zs ll sp < 2(2j + ep ky'. 2 1 k ) < 4j + y '. . cp

3 Existence and Uniqueness of Solutions

We conclude equation dominated by ( using * ))

x i+1

Next we iterate from

(

253

(i + 1) has a unique solution in -SP with norm xi +1

< ui + y i < 4j + 2yi < 8j + 2 1 + kz)xi .

=

(

-

i 0 to £ 1 to conclude that (2 2kz/ - 1 } . xe < 8 { + 1 + 2kz J . -

J+ Finally, since zTt - , we have seen that the equation and moreover has a unique solution in xe + J - JTt - . Therefore }. < xf + 2j, and hence C(k, < 2 + 8{ D

Z=

SP,

I X I sp

X=

X = f F(X) s-dZs

(2��;�:-l

Z)

= (Z1, . . . , zd),

Given a vector of semimartingales Z Zo = 0 IJ]l, 1 < i < n, and operators which are junctional Lipschitz ( 1 < i < n, 1 < j < the system of equations

Theorem 7. processes Ji E

Fj

d),

(1 < i < n) has a solution in JI]ln , and it is unique. Moreover if (Ji) i < n is a vector of semimartingales, then so is (X i ) i
J and vided we take to be matrix-valued and to be vector-valued. Hence we give here the proof for n 1. Thus we will consider the equation

F

X,

=d=

Z

( ) < k < a.s. Also, by considering the equa­ Xt = (Jt + 1t F(O) s-dZs ) + 1t G(X)s- dZ8, where G(X) = F(X) - F(O), it suffices to consider the case where F(O) = 0. We also need Lemmas 1 and 2 only for p = 2. In this case c2 JS. E S2 and such Let T be an arbitrarily large stopping time such that that zr- E S(4)sk ). Then by Lemma 2 there exists a unique solution in S2 of X (T) t = f{- + 1t F(X(T))s_ dz'{- . By the uniqueness in S2 one has, for R > T, that X( R) - X(T ) - , and therefore we can define a process X on [0, oo ) by X = X(T) on [0, T ) . Thus we have existence . Assume that maxi,J supt Ki'1 tion

w

oo

f2 x

J

= T-

T

=

T

254

V Stochastic Differential Equations

Suppose next Y is another solution. Let S be arbitrarily large such that (X Y) 8- 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( fsk ), we know that x R - = yR - by the uniqueness established in Lemma2 2. Thus X = Y, and we have uniqueness. We have assumed that maxi ,j supt K;'1 (w) < k < oo a.s. By proving existence and uniqueness on [0, to] , for to fixed, we can reduce the Lips­ chitz processes K; ,j to the random constants K;�1(w), which we replace with K (w) = maxi ,j Ki�1 (w). Thus without loss of generality we can assume we have a Lipschitz constant K(w) < oo a.s. Then we can choose a constant c such that P(K < c) > 0. Let Dn = {K < c + n}, each n = 1, 2, 3, . . . . D efine a new probability Pn by Pn ( A ) = P(A n D n ) / P(Dn ), 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 Dm 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 Dm, a.s. (dPm) · Define Yt =

()()

2: Yt 1 {nn \rln-d n= l and we have Y = yn a.s. (dPn ) on Dn , and hence also a.s. (dP) on Dn , each n. Since D = U� (O n \ Dn- d a.s. (dP), we have that on Dn : t Yt = Jt + F(Yn ) 8 - dZ8 t = Jt + F(Y)s - dZs 1

lo 1

a.s.

.

D (dP), for each n. This completes the proof. 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 = oo. 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 (cf., 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 existence of a solution), namely

a fortiori establishing the

Xt = Xo exp{ Zt - � [Z, Z]t } O<s
The uniqueness of this solution is a consequence of Theorem 7, or of Theo­ rem 6. A traditional way to show the existence and uniqueness of solutions of ordinary differential equations is the Picard iteration method. One might well wonder if Picard-type iterations converge in the case of stochastic differential equations. As it turns out, the following theorem is quite useful. Theorem 8. Let the hypotheses of Theorem 7 be satisfied, and let ( i Hi be processes in lJ]l ( 1 < i < n) . Define inductively

X0 ) =

in addition

and let X be the solution of

Xf = Jf + L 10 F} (X)s- dZt (1 < i < n). d

j =l

t

Then xm converges to X in ucp. Proof. We give the proof for d = n =m1. It is easy to see that if limm__,oo xm = prelocally in S2 , then limm __,oo x = X in ucp; in any event this is proved Xin Theorem 12 in Sect. 4. Thus we will show that limm__,oo xm = X prelocally in S2 . We first assume SUPt Kt < a < 00 a.s. Without loss of generality we ca;-assume Z E S(o:), with o: = Jsa , and that J E S2 . Let 0 = To < T1 < 2 m - are · · · < Tk be the stopping times that a-slice Z. Then ( X ) T' - and xr, in S2 by Lemma 1. Then by Theorems 2 and

3. Therefore

256

V

Stochastic Differential Equations

ll (xm+ 1 - x f1 - lls2

=

so that limm --+oo Since

� II (Xm - x f1 - lls2 2 < 2� II (X1 - X) T1 - I Is2 <

II (Xm + 1 - X) T1 - II s2

=

= 0. We next analyze the jump at T1 .

we have

II (X m+ 1 - X f1 ll s2 < II (Xm + 1 - X) T1 - IIs2 + azii (Xm - X f1 - ll s2 , where z IIZIIHoo < oo. Therefore

-

=

-

-

II (Xm + 1 - X) T1 11s2 < 2�- 1 (1 + az) II (X 1 - X) T1 - I Is2 . Next suppose we know that

I I (X m+ 1 - X ft lls2 < where 'Y II (X 1 - X) Tk - ll 82 . Then

=

(m + 1)::�� + az)e'Y

= (X;" + 1 - Xt) Tt + l{tr£ (F(Xm)s- - F(X)s - )dz�+ 1

(X;"+ l - Xt ft + 1 -

=

zH 1 (Z - zTt f£ + 1 - . Therefore by iterating on £, 0 < £ < k, II (X m+ l - X) T£+ 1 - IIs2 < ll(xm+ 1 - x f£ ll s2 + av'so:ii (Xm - x f£+ 1 - lls2 + az)e + < ( +1 (Xm - x fn 1 - ll s2 av'so:II 1 ( + 1 )£- 1 ( 1 + az)e'Y . < 2m - 1 Note that the above expression tends to 0 as tends to oo . Therefore xm tends to X prelocally in by a (finite) induction, and hence limm--+ oo xm X in ucp. It remains to remove the assumption that supt Kt < a < oo a.s. Fix a t < oo; 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 > 0 be such that P(Kt < c) > 0. Define f2n {w : Kt(w) < + } and Pn (A) P(A i f2n ) · Then Pn « P, and under Pn , limm--+oo xm X in ucp on [0, t]. For > 0, choose N such that > N implies P(f2�) < Then where

-

m

-

)�=�( 1

m

m

S2

=

c

hence limm--+oo P( (Xm must be zero.

n

=

n

=

t: .

- X);

> o) <

t:.

Since

t:

>

,

=

t:

0 was arbitrary, the limit D

4 Stability of Stochastic Differential Equations 4 Stability of Stochastic Differential Equations

257

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 Jn , J are in lDl, zn , Z are semimartingales, and pn , F are functional Lipschitz, with Lipschitz processes Kn , K, respectively. We will assume that the Lipschitz processes Kn , 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 Z0 = 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 lDl, there exists a non-random constant c < oo such that F ( H) * < c. Theorem 9. Let J, Jn E lDl; Z, zn be semimartingales; and F, pn be func­ tional Lipschitz with constants K , Kn , respectively. Assume that (i) J, r are in (resp. H ) and limn _,oo Jn = J in (resp. H2 ) ; (ii) p n are all bounded by the same constant c, and limn_,oo Fn (X) = F(X)

2

82

82

in 82, where X is the solution of ( *); and (iii) max(supn Kn , K) < a < oo a.s. (a not random), Z E S ( )s-J , (Zn ) n > l 2 n 2 are in H and limn _,oo z = Z in H2 . 2 H2 ), where x n is the solution of (*n) Then limn --->oo x n = X in 82 (resp. in and X is the solution of ( *). Proof. We use the notation H · Zt to denote J� H8 dZ8 , and H · Z to denote the process (H · Zt)t > O · We begin by supposing that J, (Jn ) n > l are in 82 and -

,

Jn converges to J in

-

82 . Then

2 S (a) is defined in Sect.

2 on page 248.

258

V

St ochast ic Differential Equations

X - xn = 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 - xn = J - r

For U E l!)l define en by

+ yn + ( Fn (X ) - Fn (Xn ) ) - · Z.

Then Gn (U) is functional Lipschitz with constant U = X - xn , and (**) becomes

a and

cn (o) =

(**)

0. Take

2 preceding Theorem 7 we have , Z) { II J - r lls2 < + IIYn lls2 C(a IIX - xn lls2 - < C (a, Z) II J - r + yn lls2 - }. Since C(a , Z) is independent of n and limn-+oo I I J - r lls2 = 0 by hypothesis, it suffices to show limn __,oo I I yn II s2 = 0. But By Lemma

< II ( F (X) - Fn (X)) - . Z lls2 + I I Fn (xn) _ . (Z - zn ) lls2 IIYn lls2 < vBIIF (X ) - Fn (X) I I s2 11 Z IIw"'

(* * * )

+ vBIIFn (Xn) - ll soo li Z - zn iiH2 by Theorem 2 and Emery's inequality (Theorem hypothesis and since

3).

Since I I ZIIH"" < oo by

lim li Z - zn iiH2 = 0, IIF (X ) - Fn(X) IIs2 = = nlim ---+ cx:> ---+ cx:> n -

again by hypothesis, we are done. Note that if we knew Jn , J E H 2 and that Jn converged to J in H2 , then We have seen already that limn-+ oo IIX - xn II S2 = 0, hence it suffices to show limn -+oo IIYn ii H 2 = 0. Proceeding as in (* * * ) we 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 (.Ft)t> o equal to .F, the Lebesgue sets on [0, 1 ] . Let ¢(t) = min(t, 1 ) , t > 0. Let f (w) > 0 and set Zf' (w) = ¢(t)f (w), w E [0, 1 ], and Zt (w) = ¢(t)f(w). n

n

4 Stability of Stochastic Differential Equations 259 Let Fn (X) = F(X) - X, and finally let JJ: = Jt = 1 , all t > 0. Thus the equations ( *n) and ( *) become respectively t

1 X�_ dZ� , t Xt = 1 + 1 X8 _ dZ8, X� = 1 +

which are elementary continuous exponential equations and have solutions

X� = exp{Zn = exp{ fn (w) ¢( t) } , Xt = exp{ Zt } = exp{ f( w ) ¢ (t) } . We can choose fn such that limn __,00 E{f� } = 0 but lim n-+oo E{f�} =/= 0 for but xn does not converge to X = 1 p > 2. Then the zn converge to 0 in for any p > 1. Indeed, limn __,00 E {!� } =/= 0 for p > 2 (since f = 0) in implies limn -+ oo E{ e tfn } =/= 1 for any t > 0. 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 as well versions in Theorem 9, but we state and prove

H2

8P,

82 H2 only the 82 versions. as

as

Let J, Jn E lDl; Z be a semimartingale; F, pn be functional Lipschitz with constants K, Kn , respectively; and let xn , X be the unique solutions of equations ( *n) and ( *), respectively. Assume that (i} Jn , J are in 82 and limn -+oo Jn = J in 82; Fn (X) = F(X) in -82, where X is the solution of (*); and (ii} lim oo -+ n (iii} max(supn Kn , K) < a < oo a.s. for a non-random constant a, and Z E S ( 2)sa ). Then limn -+oo xn = X in -82 where xn is the solution of ( *n) and X is the solution of ( *). Proof. Let xn and X be the solutions of equations ( *n) and ( *), respectively. Theorem 10.

•

Then

We let yn = (F(X) - Fn (X))- Z, and we define a new functional Lipschitz operator en by ·

Then

cn (O) = 0. If we set U = X - xn, we obtain the equation = J - r + yn + cn ( U) - . z.

u

260

V Stochastic Differential Equations

Since Z E H00 , by Emery's inequality (Theorem 3) we have yn ----> 0 in H2 and hence also in 82 (Theorem 2 ). In particular II Yn ll s2 < oo, and therefore by Lemma 2 in Sect. 3 we have ,

11 U IIs 2 < C( a , Z)IIJ - r + yn l l s2 , -

-

where C( a, Z) is independent of n, and where the right side tends to zero as n ----> oo. Since U = X - x n , we are done. o

We now wish to localize the results of Theorems 9 and 10 so that they hold for general semimartingales and exogenous processes Jn , 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 Mn are said to converge locally (resp. prelocally) in 8P (resp. HP) to M if Mn, M are in 8P (resp. HP) and if there exists a sequence of stopping times Tk increasing tooo a.s. such that limn_, 00 II ( Mn ­ M ) Tk l{ Tk >O} lisp = 0 (resp. limn _,oo II( Mn - Mfk - l l sp = 0) for each k > 1 (resp. 8P replaced by HP).

Let J, Jn E lDl; Z be a semimartingale ( Z0 = 0); and F, pn be functional Lipschitz with Lipschitz processes K, Kn , respectively. Let xn, X be solutions respectively of Theorem 11.

X;' = J;' +

1t Fn (Xn )8_dZ"'

Assume that {i) Jn converge to J prelocally in 82 ; (ii} Fn(X) converges to F(X) prelocally in 82 where X is the solution of (*); and (iii) max (supn Kn , K) < a < oo a.s. (a not random). Then limn _, oo xn = X prelocally in 82 where xn 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( Jsa ) by Theorem 5, and that 2 2 Jn converges to J in 8 and F (X n ) converges to F( X) in 82 , 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 Let J n , J E lDl; Z be a semimartingale (Zo = 0); and F,

261

Corollary. pn be functional Lipschitz with Lipschitz processes K, Kn , respectively. Let X , x n be as in Theorem 11. Assume that (i) Jn converges to J in ucp, (ii) Fn (X) converges to F(X) in ucp, and (iii) max(supn Kn , K) < a < oo a.s. (a not random) . n = X in ucp. Then lim x oo n--+ 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 subsequence n' such that infn, d ( Xn , 0) > 0. Therefore no sub-subsequence (X n ) can converge to 0 in ucp, and hence x n" cannot converge to 0 prelocally in 82 as well. Therefore to establish the result we need to show only that for any subsequence n ', there exists a further subsequence n" such that x n" con­ verges prelocally to 0 in 82 . This is the content of Theorem 12 which follows, 0 so the proof is complete. "

Let Hn , H E lDl. For 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 82 . Proof We first show the necessity. Without loss of generality, we assume that by iteration a decreasing sequence of subsets (N k ) of HN =={1,0. 2,We3, .construct . . }, such that lim sup IH� I = 0 a.s. --+oo n Theorem 12.

n ENk O < s
By Cantor's diagonalization procedure we can find an infinite subset N' of N such that = 0 a.s. , lim sup n -E-->Noo' O <s 0. By replacing N with N' we can assume without loss of generality that tends to 0 uniformly on compacts, almost surely. We next define

I H;' I

Hn

Tn 8

n

=

inf{t > 0 = inf

T > m n m.

: I Hf l > 1},

Then Tn and 8n are stopping times and the 8n increase to oo a.s. Indeed, for each k there exists N (w) such that for n > N (w), SUPo-<s k and 8N(w) (w) > k. Next, define -

Lm =

(Hm ) ( Sn!l.n) - .

I H;'(w) l

262

V Stochastic Differential Equations

Then which tends to

(L m )* < o<s< sup [H;'[,

n

0 a.s. Moreover, when > n, (Lm )* < sup [H;'[ < sup [H;' [ < 1. m

O <s< Sn

O<s< Tm

Hence by the Dominated Convergence Theorem we have that (Lm )* tends to 0 in L2, which implies the result. For the sufficiency, suppose that Hn does not converge to H in ucp. Then there exist t0 > 0, 8 > 0, c: > 0, and a subsequence such that P{(Hn' H);0 > 8 } > c: for each n'. Then II (Hn ' - H);0 [[ £2 > 8JE for all n'. Let Tk tend to oo a.s. such that (Hn' ) tends to 0 in 82 , each k. Then there exists K > 0 such that P(T k < t 0 ) < 8f for all k > K. Hence -

We conclude

u�.

H Tk -

n'

8JE < 8f , a contradiction. Whence Hn converges to H

m

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 8P and prelocal HP convergence are not topological in the usual sense. If they were, then one would 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 82 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 such that limn' --> oo x n' = X, prelocally in 82 . However we saw in the comment that x n does not converge to X in 82 • It is still a priori possible that x n converges to X prelocally in 8 2 , however. 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 fn on [0, 1 ] such that limn -->oo J01 fn (x) 2 dx = 0 and lim supn--+ oo JA (fn (x))Pdx = +oo for all p > 2 and all Lebesgue sets A with strictly positive Lebesgue measure. In conclusion, this counterexample gives a sequence of semimartingales x n such that every subsequence has a sub-subsequence converging prelocally in 8P, but the sequence itself does not converge prelocally in 8P , (1 < p < oo) .

n'

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 Xn 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

t

1 Fn (xn )sdWs + 1 t Gn (Xn )8ds , t t Xt = Jt + lo F(X) s dWs + 1 G(X)8ds

Xf = Jf +

where W is a standard Wiener process. If ( Jn - J)t converges to 0 in L 2 , each t > 0, and if pn, en , F, G are all functional Lipschitz with constant K < oo and are such that (Fn( X ) - F(X))t and (Gn (X) - G(X))t converge to 0 in L 2 , each t > 0, then (X n - X)t converges to 0 in L2 as well, each t > 0. Note that we require only that F n (X ) and cn (X) converge respectively to F( X) and G(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 Jn , J E ]])) ; Z, zn be semimartingales (Z0 = Zo = 0 a.s.);

and F, pn be functional Lipschitz with Lipschitz processes K, Kn , respectively. Let x n , X be solutions of ( *n) and ( *), respectively. Assume that (i) Jn converges to J pre locally in 82 ; (ii} Fn (X) converges to F(X) prelocally in 82 , and the coefficients Fn , F are all bounded by c < oo; (iii) zn converges to Z prelocally in H2 ; and {iv) max(sup n Kn , K) < a < oo a.s. (a not random). Then lim xn = X prelocally in -82 • -HX> n Proof By stopping at T- for an arbitrarily large stopping time T we can assume without loss that Z E S ( }s) by Theorem 5, and that Jn converges 2 n 2 to J in 8 , p (X) converges in 82 to F( X), and zn converges to Z in H2 , 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 ]]))

V

264

Stochastic Differential Equations

r(Y ) =

2- n E{ 1 /\ sup l}, IYt L t O < O

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) = Hsup r(H · X) I I
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 < oo, let xn 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 xn 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 IIXII H2 and sup i H I i i H · Xll s2 in the corollary to Theorem 24 in Sect. 2 of that 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 Jn , J E lDl; zn , Z be semimartingales (Z[; = Z0 = 0); and pn , F be functional Lipschitz with Lipschitz processes K, Kn , respectively. Let x n , X be solutions of (*n) and (*), respectively. Assume that (i) Jn converges to J in ucp, (ii) pn (X) converges to F (X) in ucp where X is the solution of ( * ) , and moreover all the coefficients pn are bounded by a random c < oo , (iii) zn converges to z in the semimartingale topology, and (iv) max(supn Kn , K) < a < oo a.s. n = X in ucp. x Then lim -HX>
n

Proof Since zn converges to Z in the semimartingale topology, by Theorem 14 there exists a subsequence n ' such that zn converges to Z prelocally in H2 . Then by passing to further subsequences if necessary, by Theorem 12 wemay assume without loss that Jn converges to J and pn(X) converges '

4 Stability of Stochastic Differential Equations 265 to both prelocally in 82 , where is the solution of ( *). Therefore, by Theorem 13, converges to prelocally in 82 for this subsequence. We

X xn X have shown that for the sequence (Xn) there is always a subsequence that converges prelocally in 82 • We conclude by Theorem 12 that xn converges to X in ucp. F(X)

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. E lDl; Theorem 15. Let be semimartingales 0); and

zn, Z

Jn, J

(Z0 = Z0 = pn, F be functional Lipschitz with Lipschitz process K, the same for all n. Let xn, X be solutions respectively of t X[' = J[' 1 Fn (Xn ) s- dZ�, +

Assume that (i) converges to in ucp; in ucp, where is the solution of (*); and {ii) converges to {iii) converges to in the semimartingale topology. Then converges to in ucp. Proof First we assume that supt Kt(w) < a < oo . We remove this hypothesis 12, it suffices to show that there exists at the end of the proof. By Theorem ' converges to a subsequence n ' such that prelocally in 82 Then by

JnFn(X) zn xn

J F(X) z X

X

xn X . Theorem 12 we can assume with loss of generality, by passing to a subsequence if necessary, that Jn converges to J and Fn(X) converges to F(X) both prelocally in 82 . Moreover by Theorem 14 we can assume without loss, again by passing to a subsequence if necessary, that zn converges to Z prelocally in H2 , and that Z S( 4v�s)· Thus all the hypotheses of Theorem 13 are satisfied except one. We do not assume that the coefficients pn are bounded. However by pre-stopping we can assume without loss that I F (X)I is uniformly bounded by a constant c < Let us introduce truncation operators Tx defined (for > 0) by Tx(Y) = min( , sup( Y)) . E

oo.

x

x

-

x

,

Then Tx is functional Lipschitz with Lipschitz constant Consider the equations

t �n = J[' 1 (Ta+c+l Fn )(Yn) s- dZ�. +

1, for each x > 0.

266

V Stochastic Differential Equations

Then, by Theorem 13, y n converges to X prelocally in 82 • By passing to yet another subsequence, if necessary, we may assume that Fn (X) tends to F(X) and yn tends to X uniformly on compacts almost surely. Next we define The stopping times 8k increase a.s. to oo. By stopping at 8k -, we have for n > k that (Yn - X) * and ( Fn(X) - F (X)) * are a.s. bounded by 1. (Note that stopping at 8k - changes Z to being in S ( }sa ) instead of S ( 4)sa ) by 2 Theorem 5.) Observe that

+

I Fn (Y n ) l < j Fn ( Yn ) - Fn ( X) j + I Fn ( X) - F ( X) j jF ( X) j < a( Yn - X) * + (Fn ( X) - F(X) ) * + F(X) * < a + 1 + c, whence (Ta+ c + l pn )(Yn) = Fn(Yn). We conclude that, for an arbitrarily large stopping time R, with Jn and zn stopped at R, y n is a solution of t Yfn = J? + Fn ( Yn )s _dZ� ,

1

which is equation ( *n ) . By the uniqueness of solutions we deduce yn = xn on [0, R). Since yn converges to X prelocally in 82 , we thus conclude xn converges to X prelocally in 82 • It remains only to remove the hypothesis that supt Kt(w) < a < oo. Since we are dealing with local convergence, it suffices to consider sups 0 be such that P( Kt < a) > 0, and define

ilm = {w : Kt (w) < a + m}.

Then ilm increase to n a.s. and as in the proof of Theorem 7 we define Pm by Pm (A) = P ( Aiilm), and define :F[' = .Ft l n , , the trace of Ft on ilm . Then Pm � P, so that by Lemma 2 preceding Theorem 25 in Chap. IV, if zn converges to Z prelocally in H2 (P), then zn converges to Z prelocally in H2 ( Pm) as well. Therefore by the first part of this proof, xn converges to X in ucp under Pm , each m > 1. Choose c > 0 and m so large that P(il�) < c . Then

n < lim Pm ((Xn - X); > 8) + P(il�) 8 P(( X); > X ) n ---+ oo n --+ oo < lim Pm(( Xn - X); > 8) + c n-HX> lim

and since c > 0 was 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 Lemma (Dominated Convergence Theorem) . Let p, q , r be given such and let that 1. 1. = 1.r , where < r < oo . Let Z be a semimartingale in Hn EP SP such that 0, < E SP, all n > Suppose lim n_, oo all Then

16.

+q (t,w).

1n IH l Y

Hq , Hf_ (w) =

1.

=

Hq , there exists a decomposition of Z, Z N + A, such that ]q (N,A) I [N, N] �2 + 1 00 l dA .II I Lq < 00 . Let en be the random variable given by 00 00 en = ( 1 (H;-_ ) 2d[N, N] s ) 1 / 2 + 1 IH-: II dAs l· The hypothesis that I Hn l < Y implies en < Y* ([N, N]�2 + 1 00 i dAs I ) a.s. However 00 00 I Y* ( [N,N]�2 + 1 j dA si ) I Lr < I Y* I Lvi i [N, N]�2 + 1 jdA sll i Lq = I Y I sv]q (N,A) < Proof.

Since

ZE

=

=

oo.

Thus en is dominated by a random variable in Lr and hence by the Dominated Convergence Theorem en tends to 0 in Lr . D 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 = {0 < <··· < nn} , we define

= To T1

Yo1{o} + l:k Yrk 1(rk,Tk+l] · Note that if Y is adapted, cadlag (i.e, Y E ill>) , then ( Yt )s > O is left continuous with right limits (and adapted, of course) . It is convenient to have version ya

of ya

E ill>, occasionally, so we define ya + L Yrk [Tk ,Tk+I) · =

k

a

1

3 Random partitions tending to the identity are defined in Chap. II, preceding

Theorem 21.

268

V

Stochastic Differential Equations

Let J E S2 , let F be process Lipschitz with Lipschitz process K < a < oo a.s. and F(O) E S2 . Let Z be a semimartingale in S( 2)s) , and let X( a) be the solution of Theorem 16.

t Xt = Jt + l F(X 1 , and an argument analogous to the proof of Theorem 10 (though a bit simpler ) shows that it suffices to show that J� en (X)8_dZ8 converges to J� F(X)8_dZ8 in S2 , where X is the solution of t Xt = Jt + 1 F(X)s - dZs. Towards this end, fix ( t, w) with t > 0, and choose > 0. Then there exists [t - 28, t). If mesh (a) < > 0 such that IXu(w) - Xt _(w) l < for all then also j x;;+(w) - Xu(w)l < 2c for all u [t - o, t). This then implies that j F(X
E

E

E

0

E

E

E

u E

E

o,

mesh (0

mesh

-->0

mesh (0

However we also know that

j en (X) - 1 = j F (X
D

4 Stability of Stochastic Differential Equations

269

Remark. In Theorem 1 6 and its corollary which follows, we have assumed that F is process Lipschitz, and not functional Lipschitz. Indeed, Theorem 1 6 is not true in general for functional Lipschitz coefficients. Let Jt = 1 {t > l} , Zt = t 1\ 2, and F( Y) = Y1 1 {t > l} · Then X, the solution of ( * ) of Theorem 15 is given by Xt = ( t 1\ 2) 1 {t > 1} , 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 = 1 {t > l } · ( Here X(a) denotes the solution to equation (*a) of Theorem 1 6.) Corollary. Let J E ill> ; F be process Lipschitz; Z be a semimartingale; and let an be a sequence of random partitions tending to the identity. Then lim -> oo X(an ) = X

n

I n ucp

where X (an ) is the solution of (*a) and X is the solution of ( *), as in Theo­ rem 16.

Proof

First assume K < a < oo, a.s. Fix t > 0 and E > 0. By Theorem 5 we can find a stopping time T such that zr - E S( 8Jsa ), and P (T < t) < E. Thus without loss of generality we can assume that Z E S( 8JsJ · By letting Sk = inf { t > o 1 Jt 1 > k}, we have that sk is a stopping time, and limk _,oo sk = oo a.s. By now stopping at Sk - we have that J is bounded, hence also in S2 , and Z E S( }sa ) . An analogous argument gives us that F(O) can be assumed 4 bounded ( and hence in S2 ) as well; hence Z E S( Js)· We now can apply 2 Theorem 16 to obtain the result. To remove the assumption that K < a < oo a.s. , we need only apply an argument like the one used at the end of the proofs of Theorems 7, 8 and 15. D :

Theorem 1 6 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 < TJ:n } , we see that the random variables partition an = { 0 = T0 < T{' < X(an ) r;: verify the relations (writing a for an , X for X( an ), Tk for TJ:) ·

·

·

Xro = Jo, Xrk + ' = Xrk Jrk+ l - Jrk + F(xa + ) rk (Zrk + l - Zrk ).

+

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

Let Z be a semimartingale and let X = £(Z ) , the stochastic exponential of Z. That is, X is the solution of t Xt = 1 + lo Xs _dZ8•

Theorem 17.

Let an be a sequence of random partitions tending to the identity. Let kn - 1 X n 11 (1 + (ZTi+ 1 - zrt )) . =

i= l

Then lim xn = X in ucp. n-->oo Proof. Let yn be the solution of yt

=

t

1 + lo ysan dZs,

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 = x n . Let an = {0 T0 < T{' < · · · < T,rJ . On (Tin , TZ+ 1 J we have =

�n

= Y,Pn + Y,Pn (zr,+. l - zrt ) = Y,Pn (1 + (Z Ti+1 - zr,n )) . '

'

'

Inducting on i down to 0 we have

�n = 11 (1 + ( ZT7+1

for Tt < t < we have that

j
-

zT? )),

T[+ 1 . Since zT7+ 1 - zrjn = 0 for all j > i when Tt < t < T[+ 1 , y n x n ' and the theorem is proved. 0 =

5 Fisk-Stratonovich Integrals and Differential Equations

In this section we extend the notion of the F isk-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] t ) t > 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 Integ rals and Differential Equations

271

n T T , + Z ' � H (H lim J H )(Jr;+ , JTt ) San (H, ) = 0J0 + i = [H, J] < Tf: . is the sequence 0 = T0 < T{' < with convergence in ucp, where n A process H in ill> is said to have finite quadratic variation if [H, H] t exists and is finite a.s. , each t > 0. 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 lim n�oo

L....t

n--+oo

_

an

· ··

consistent. Notation. For of finite quadratic variation we let denote the con­ tinuous part of the ( non-decreasing paths of Thus,

H

[H, H]c [H, H]. ) [H, H]t = [H, H] f + O<s
The next definition extends the definition of the Fisk-Stratonovich integral given in Chap. II, Sect. 7. Definition. Let E ill> , be a semimartingale, and assume exists. denoted J; The Fisk-Stratonovich integral of with respect to is defined to be

H

dX8,

X

H

X,

[H, X]

H8_

o

rJo t Hs- odXs - Jor t H8_dX8 + 21 [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. be an Theorem 18 (Generalized Ito's Formula) . Let X = n-tuple of semimartingales, and let f JR + x 0 x lRn --> lR be such that

(X l , . . . , Xn)

:

(i} there exists an adapted FV process A and a function g such that

t

f(t,w,x) = lo g(s,w,x)dA8, (s, w) g(s, w, x) is an adapted, jointly measurable process for each and J; supxE K j g ( s , w , x) l l dA s l < oo a.s. for compact sets K. t---+

x,

V Stochastic Differential Equations (ii} the function g of (i} is C2 in x uniformly in s on compacts. That is,

272

n

xi ) s,w,x))( ( sup{ Y x i i s
X

=

--->

=

=

Then

f(t,w, Xt) t t f(O,w,Xo) + 10 g(s,w, X8)dA8 + L= 10 fx;(s-,w, X8_)dX; 1 i . 1 lot )d[ X ',X j(s-,w, f 8_) + L L {f(s,w, 8)-f(s-,w, ] + Xs X X � ; 1 x x O<s
=

2

n

.

n

=

=

=

=

=

u

,

as

5 Fisk-Stratonovich Integrals and Differential Equations

integral since A is an FV process. Expanding the integrand for fixed by the Mean Value Theorem yields

273

(u,w)

g(u,w, Xt k+l ) = g(u,w,Xu ) + gx (u,w,Xu )(Xtq1 - Xu) where Xu is in between Xu and Xtk+ l " Therefore A

A

X t]

and since is of finite variation and is right continuous, the second sum tends to zero as the mesh of the partitions tends to zero. Therefore letting 1rn denote a sequence of partitions of [0, with limn -we mesh(7rn ) 0,

=

a.s.

( ) namely t ,w,Xtk). f( f(tk, k ,Xtk+J :2: W k=O Here we proceed analogously to the proof of Theorem 32 of Chap. II. Given E > 0, t > 0, let A(c, t) be a set of jumps of X that has a.s. a finite number of times and let B = B(c,t ) be such that L s EB (L\X8)2 < c2 , where A U B exhaust the jumps of X on (0, t]. Then ) ,Xtk f(tk, W l:J(tk,w,Xtk+1) k=O t (t ) f(tk,W, Xt ) ,Xtk k, (t k, ,XtJ + L J ) f( k,W,Xtk W + W LJ + l k l k, B k, A Next consider the second term on the right side of

* ,

m- 1

s,

m-1

=

By Taylor's formula, and letting

L\kX denote Xtk+l - Xtk ,

274

V

Stochastic Differential Equations

k,B

2 X (t tk,W,X Lf + Lf k,W,Xtk)(!:l.kX) tk)!:l.k ( � xx x k k 1 "' 2 t X X X tk, - L.. w, + f k, w, Xt ( ( tk )!:l. k ) J k) (!:l.k xx x 2 k R t ,Xt k,w,Xt + l: ( 1) k · + k k,B By Theorems 21 and 30 of Chap. II, the first sums on the right above con­ to J� fx (s- , w, Xs - )dXs and � J� fxx (s -, w, Xs - )d[X , X]8, re­ verge in spectively. The third sum converges a.s. to - :2: Ux ( W , Xs - )!:l.Xs + � fxx (s - , w, Xs - )(!:l.X8)2}. By condition (ii) on the function g, we have =

,

A

ucp

s EA

where

S-,

rt(w, c:+) = lim sup rt(w,o). Next we let tend to 0; then rt(w,c:+)[X,X]t tends to 0 a.s., and finally combining the two series indexed by A we see that :2: {f(s-,w,Xs) - f(s- ,w,Xs-) - fx (s-,w,Xs - )!:l.Xs - 21 fxx (s-,w,X8 - )(!:l.X8) 2 } E

o.J.c:

sEA(c:, t )

tends to the series

L {f(s - ,w,Xs) - f(s-,w,Xs - ) - fx (s-,w,Xs - )D.Xs ( I:o< s
which is easily seen to be an absolutely convergent series cf., the proof of The­ into orem of Chap. II ) . Incorporating yields the term � = and We further note that is absolutely convergent, the theorem is proved. since < k for < t is removed as it was in Theorem The assumption that of Chap. II. 0

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 = (X 1 , . . . , X n ) is an n-tuple of semimartingales. Also, an important special case is when the process A8 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 = (X 1 , . . . , X d ) be a vector of semimartingales and let j : �+ X f! X �d --> � be such that (i) there exists an adapted FV process A and a function g such that ·,

_

t

f(t, w, x ) = lo g(s, w, x)dA8 where ( s, ) g ( s, ) is an adapted and jointly measurable process for each (ii} for each compact set K, J� supxE K lg ( s , , ) l l dA l < oo a.s. (iii) fxi exists and is continuous in and fxi (t , , ) = J� gx,(s,w,x)dA8• Let yt = f(t, X 1 , . . . , xd ). Then Y is an adapted process of finite quadratic variation, and moreover x.

w

�---->

w, x

w x

x

s

w x

w,

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,

Proof.

where

an = {0 = T0 < T{' < . . · < lkn }. Since fk and their first partials converge to f and its first partials uni­ formly, we have fk - !£ is Lipschitz with constant Eke , which tends to 0 as k, £ tend to oo . Therefore, letting denote an for a given n and writing fk (X) for fk (t, Xt ) , we have w,

T

276

V Stochastic Differential Equations

IST ( !k (X)) - ST ( h(X)) \ < ST (( /k - h)(X)) + 2{ST ( !k (X))ST (( !k - /£ )(X ))} 1 1 2 (*) < (c�" + 2cck£)Sr (X) by the Lipschitz properties. Since fk (X ) and f£(X) are semimartingales, we know ( by Theorem 22 of Chap. II ) that San ( /k (X)) and Sa n ( !£ (X )) converge in ucp respectively to [fk (X), fk (X)] and [h(X), /£ (X)] . By restricting our attention to an interval ( s, t], we then have from ( *) that \ { [fk (X), /k (X)]t - [fk (X), fk (X)]s} - { [f£(X ), !£ (X ) ]t - [ !£ (X ), fl' (X)]s}l < (ck£ + 2cc k£){ [X, X] t - [X, X]s }· Since this is true for all 0 < s < t < oo , we deduce that Therefore d [fk(X), fk (X)] is a Cauchy sequence of random measures, con­ verging in total variation norm on (0, oo ) , 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 ( D.fk (X)) 2 = A[/k (X), fk (X)] .

Therefore

[fk (X), fk (X)]t =

t

1 c::: (s-, w,Xs- )2d [X, X]� + O<2:s
and since we have a.s. convergence in total variation norm, we can pass to the limit to obtain

vt =

t

1 J'(Xs-)2d [X, X]� + O<2:s
Our identification of the limit removes the dependence on the representation of Y by f, because

\ San ( f (X))t - vt\ < \ San (f(X))t - San ( fk (X)) t \ + \vt - [ /k (X), /k (X)]t \ + \ San ( /k (X))t - [/k (X ), /k (X)]t \ , 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

(f(X)) t [Y, Y] t d f x, gm d f x g x g (x) J J m m m m (w)}. m f(t,w,gm (Xt )), T (w) � 0 : � yt(w) ym [Ym , ym Y ] t (w) [Y, Y] t (w), t Tm((w). d [Y m , Y m ] t t = l ��(s-,w,gm (X8 _ ))g'm (X8_)d[X ,X] � + L f1J (s,w,gm (X8 ))2 0 O<s< t which, on [0, Tm ), is equal to

= third term to make it small in probability, and we have limn ---.oo San and this completes the proof for vt , in probability. Therefore vt = and its first partials bounded functions of and = 1. The proof for general : JR.d is exactly analogous. For general satisfying the hypotheses, let < m, and has compact support. = if JR.d be C00 such that let = inf { t > = Defining :f. Then increase to oo a.s. Also, since the quadratic variation is a path = property, for all < Therefore is of finite quadratic variation, and since by the preceding for = 1) -+

whence the result.

0

Y

Y

Note that the subspace of processes that have a representation = 1, . . . is a finite-dimensional semimartin­ where X = (X 1 gale is not fixed) and E 1 is a vector subspace of the space of processes of finite quadratic variation. As an immediate consequence of Theorem 19 we have the following extension of Theorem 34 of Chap. II.

f (X (d , Xd),

. . . Xd) , , f C,

Let X be a semimartingale and let f be C2. Then t f(Xt) -J(Xo) = 10+ f'(Xs_)odXs + O<s< L t {f(X.) -f(Xs_) -J'(Xs_) f1Xs}. Proof Note that f' is C 1 , so that f' (X) is in the domain of definition of the F-S integral by Theorem 19. Also by definition we have t{ f (Xs ) odXs = {t J'(Xs )dXs + � [ X]� . !'(X), ' 2 Theorem 20.

h+

By Theorem

h+

19 and polarization we know that

t [J' (X),X]� = 1 f'' (X8 _ )d[X,X] � ,

and thus the result follows by Ito's formula (Theorem

32, Chap. II ) .

0

Theorem 20 has an obvious generalization to the multi-dimensional case. We omit the proof which is an analogous consequence of Theorem 33 of Chap. II.

V

278

Stochastic Differential Equations

Let X = (X l , . . . ,X n ) 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:

Theorem 21. ---+

:

f(Xt ) - f(Xo) t of ) . n n o = L 1 OX (Xs - o dX� + L {f(Xs) - J(Xs - ) - L oxfi (Xs - )�X�. } . O<s
parts formula.

21,

we have the

Stratonovich integration by

Corollary 1 (Stratonovich Integration by Parts Theorem) . Let X and Y be semimartingales. Then

Proof.

Let

f(x, y)

=

xy and apply Theorem 21.

0

X and Y be semimartingales, with at least one of X or Y t XtYt - XoYo = lto+ Xs - odYs + l{o+ Ys - o dX • Recall that since Xo - = 0 by convention for a dtdlag process X, we did not really need to write J;+ ; the formula also holds for J;. Corollary 2. Let continuous. Then

8

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 acceptable if

:

JR. +

X

nX

JR_d

-4

JR. is said to be Fisk-Stratonovich

A t j(t,w, x) = lo g(s,w,x)dAs where ( s, w) g( s, w, x) is an adapted, jointly measurable process for each x; ( ii ) for each compact set K, J; supx E K i g (s,w, x) l l dAsl oo a.s. ; (iii) ! is C 1 and (i ) there exists an adapted

FV process and a function g such that

�----'

<

xi

5 Fisk-Stratonovich Integrals and Differential Equations

279

(iv) for each fixed (t, w ) , the functions x f.---* f(t, w, x) and x f.---* ( %f, f)(t, w , x ) are all Lipschitz continuous with Lipschitz constant K(w), K < oo 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 = (Z 1 , , z k ) , semi­ martingales Ji ( 1 < i < d), and F-S acceptable junctions fj (1 < i < d, 1 < j < k) , then the system of equations •

x;

=

.

•

k t Ji + L fj ( s w X - ) o dZ1 0

j

1 =l

-,

,

s

has a unique semimartingale solution. Moreover the solution X of ( ) is also the (unique) solution of *

( ) **

We note that equation ( ) has a unique solution as a trivial conse­ quence of Theorem 7. Since X is a d-dimensional semimartingale, we know that fj (s-, · X8_ ) 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

Proof

**

,

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 D variation, the terms involving [Jm , Z1Jc disappear. *

**

**

*

4 5

,

By ( :f, f)(t , w, x) we mean the usual product of functions :f, (t, w, x) f(t, w, x) . Since we are taking the continuous parts of the quadratic variations, we need not write f(s- , w , Xs- ) , 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.

the equation

is given by

Let Z be a semimartingale,

Xt = Xo exp{Zt }

IT

O< s
Z0 =

0. The unique solution of

( 1 + �Z8 )e- 6.Z. ,

and it is called the Fisk-Stratonovich exponential, X0£ ( Z) . Proof By Theorem 22 the equation above is equivalent to F-s

1 1

1

t

1 t Xs_d[Z, Z]� X8_ dZs + Xt = Xo + 2 0 0 t 1 = Xo + Xs_ d(Zs + - [Z, Z]� ) .

2

0

Therefore

EF-s(Z) = E(Z +

1

2 [Z, z]c)

�

= E(Z)£( [Z, zn

1 2

= f(Z) exp { [Z, z]c},

where the second equality is by Theorem 38 of Chap. II. Using the formula for the stochastic exponential established in Theorem 37 of Chap. II, the result 0 follows. The most interesting case is when the semimartingale is continuous. Corollary 1. Let Z be a continuous semimartingale, Z0 = 0. Then the unique solution, £F-S (Z), of the exponential equation

is given by EF-s(Z) t = exp { Zt } . Corollary 2. Let E be a Brownian motion with Eo = solution of t Xs o dEs Xt = Xo + is given by Xt = Xo exp { Et } .

lo

0. Then the unique

5 Fisk-Stratonovich Integrals and Differential Equations

28 1

The simplicity gained by using the F-S integral can be surprisingly helpful. As an example let Z = (Z 1 , . . . , zn ) be an n-dimensional continuous semi­ martingale and let = (x 1 , , x n )' be a column vector in JR. n . (Here denotes transpose. ) Let I be the identity n x n matrix and define

x

.

.

1

•

xx'

a(x) = I - l x l

2

.

represents projection onto the hyperplane normal to x.) a(x) x' a (x) = 0. We want to study the system of F-S differential equat Xt = Xo + 1 a(Xs ) dZ8, 1 < i < n.

(The matrix Note that tions

o

where X = (X 1 , . . , X n ) ' and Z = (Z 1 , . . . , zn )'. Theorem 24. The solution X of equation ( * * * ) above it always stays on the sphere of center 0 and radius .

Proof

exists, is unique, and

l xo l ·

a

Since is singular at the origin, we need to modify it slightly. Let be a c= function equal to a( ) outside of a ball centered at the origin, Let Y be the solution of such that tf-

x

x0 No.

Yt = Xo +

g(x) N0,

t

1 g(Ys) o dZ8•

Y exists and is unique by Theorem 22. Let T = inf { t > 0 : yt E No} . Y0 = x0 and Y is continuous, P(T > 0) = 1 . However for s < T, g(Ys) = a(Y8). Therefore it suffices to show that the function t II Yt l l is constant, since II Yo I = l xo I · This would imply that Y always stays on the sphere of center 0 and radius l xo l , and hence P(T = oo ) = 1 and g(Y) = a(Y) always. To this end, let f(x) = I::� 1 (xi )2, where x = (xi, . . . , x n ) ' Then it suffices to show that df(Yt) = 0, (t > 0). Note that f is C2• We have Then Since

�----'

.

that for t < T,

df(Yt) = 2Y� o dY = 2Y� · a(Yt) dZt = 0. Therefore I Yt l = llxo ll for t < T and thus by continuity T = oo a.s. Corollary. Let B = (B\ . , En )' be n-dimensional Brownian motion, o

a (x)

=

.

I - ��2 , and let X be the solution of .

.

'

Then X is a Brownian motion on the sphere of radius

I x0 II·

D

let

282

V

Stochastic Differential Equations

In Theorem 36 of Sect. 6 we show that the solution X is a diffusion. By Proof Theorem 24 we know that X always stays on the sphere of center 0 and radius ll xo 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)

o

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 dX = Ua( X) dB = Ua(X)U'U dB = a( UX) d( U B), o

o

o

o

and we are done.

D

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 t t + Xo +

Xt =

lo f(X8) dZs lo g(X8)ds, o

Z is a continuous semimartingale, Z0 = 0. (Note that J; g(Xs) ds = so we have not included Ito's circle in this term.) Assume that f J; Cg(Xs)ds, is 2 and that j,g, and ff ' are all Lipschitz continuous. Let u = u(x, z) be the unique solution of au (x , z) = f(u(x, z)) oz u(x, 0) = x. Then tz �� = f' (u(x, z)) ��, and �� (x, 0 ) = 1 , from which we conclude that z ��(x,z) = exp{fo f'(u(x,v))dv }. Let Y = (Yt) t >o be the solution of {Zs Yt = xo + Jto exp{- Jo f'(u(Y8,v))dv}g(u(Y8, Z8))ds g (x, Zs)) is Lipschitz, this would which we assume exists. For example if �� a (x, Zs) where

suffice.

o

x

5 Fisk-Stratonovich Integrals and Differential Equations

283

With the notation and hypotheses given above, the solution X of (*4) is given by

Theorem 25 .

Proof.

Using the F-S calculus we have

t 8u

t 8u

u(yt, Zt) u( xo , 0) + Jo ox (Ys, Zs) o dYs + Jro oz (Ys, Zs) o dZs Zs t t { Xo + J exp { j f'(u(Ys, v))dv} o dYs + J f(u(Y8, Z8)) o dZ8 • o o =

=

Since

0

dYs ds

= exp{ -

{ Zs f (u(Ys, v))dv}g( u(Y8 , Z8)), I

Jo

we deduce

By the uniqueness of the solution, we conclude that

Xt = u(Yt, Zt) ·

0

We consider the special case of a simple Stratonovich equation driven by a (one dimensional) Brownian motion B. As a corollary of Theorem 25 we obtain that the simple Stratonovich equation Xt has a solution

=

Xo +

t

1 f(Xs ) o dEs

Xt = h - 1 (Bt + h(Xo)), where h(x )

=

1 r Jo 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

dXt

=

-

1 a Xtdt + a yI 2 2

has solution Xt = sin (aBt + arcsin(Xo)).

1

-

Xt dBt 2

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 Xt = (XJ - m - a(m - 1)Bt) ! Example. Our last example can also have explosions: ,

= .

dXt = Xt (1 + X'f)dt + (1 + Xt2 )dBt has the solution Xt = tan(t + Bt + 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 semimartingale. Assume [H, X] exists. Let an = {0 = T0 < Tf < · · · < Tk:, } be a sequence of random partitions tending to the identity. If H and X have no jumps in flH8ilX8 = 0, all t > 0), then common (i. e.,

Lo<s + Hrr> ) (XTn hm i

'

>+I

i+t

Tn X - ;)

equals the F-S integral J� H8_ dX8 in ucp. Proof It follows easily from the definition of [H, X] at the beginning of this o

section that tl[H, X]t = flHtflXt and limn---. oo I.:(Hrz+ , - Hrt ) (XT;''t , X Tn ) = [H, X] - H0X0 . Thus if H and X have no jumps in common we conclude [H, X] = [H, X]c + H0X0 . Observing that ;

the result follows from Theorems 21 and 22 of Chap. II. Corollary 1. If either H or X in Theorem 2 1 is continuous, then

where H0_ = 0, in ucp.

D

5 Fisk-Stratonovich Integrals and Differential Equations

285

X

Corollary 2. Let and Y be continuous semimartingales, and let an = {0 = T0 < TJ' < · · · < TJ:;, } be a sequence of random partitions tending to the identity. Then hm .

n-HXl

1 -(Y L 2 r:> + Yr:•>+ 1 )(XT•:'+ 1 - XT:' ) = 1 " Ys - o dX8, •

'

.

'

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). D

Let H be cadlag, adapted, of finite quadratic variation, and suppose :L:o <s< t I�Hsl < oo a. s., each t > 0. Let X be a semimartingale, and let an = [Tr }o< i
Theorem 27.

with convergence in ucp. Proof First observe that fit = Ht - :L:o <s< t �Hs defines a continuous process of finite quadratic variation and that [if, X] = [H, X] c + H0X0 . Next we note

that 1 2

1 (H - H = H n H H �Hs) + ( r;- + r,f. 1 -Tn
-

1.

1.

-

1.

2

Therefore by Theorems 2 1 and 22 of Chap. II, or directly from the definition as in the proof of Theorem 26, we have

n A A . n n n . 1 r T = hm L Hr n(X ' + 1 - X ' ) + hm - L (Hr"-n+ 1 - Hr"'!' )(X T' + 1 - XT' ) 2 • 1 = 1 Hs_dXs + -([H ,XJ. + HoXo) 2 O+ 1 = . H8_ o dX8 + -[H,X]� = " Hs - o dX8, where Ho - = Ho, 2 1 1 n-+oo

i

"'

n-+oo

i

A

0

and the result follows.

0

D

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

Jt =

L �Hs l { l.:lHs l >.s} ·

O< s < t

Then J"' is also a cadlag, adapted process, and it has paths of finite variation on compacts. Theorem 28. Let H be cadlag, adapted, of finite quadratic variation, and let

X be a semimartingale. Assume [H, X] exists. Let an = {�n } be a sequence of random partitions tending to the identity. Let ]"' be as defined above. Then

with convergence in ucp. Proof. Let H"' = H - J"'. Then the jumps of H"' are bounded by

c.

Moreover,

Therefore the first limit at t > 0 is

0 gives the result. That is, since [H"' H "'] t1 1 2 [X ' X] t1 / 2 � l ll Hs"' !lX I "'"'

and letting c tend to

$

and since

0.

<

_

'

< 00

'

[H"', H"'] t = "Eo < s < t (� H;) 2 , we have lim.s -+0 "E o < s < t I�H; �Xs I =

D

The next result is useful, for example, for approximating integrals of the form J; f(Bs) dB8, where B is a Brownian motion. It can be generalized to Y a continuous, adapted process of finite quadratic variation, but we do not do so here. o

Let X be a semimartingale and Y a continuous semimartin­ gale, and let f be C 1 . Let fJ be a probability measure on [0, l] let a = J AfJ(d>.) , and let an = {Trh>o be a sequence of random partitions tending to the iden­ tity. Then

Theorem 29.

287

5 Fisk-Stratonovich Integrals and Differential Equations

1 f(Yrn + >.(Yr:> - Yrn ) )J.-t(d>.) (Xrn - Xr'n ) 1 n .o......+ oo � = {" f(Ys ) dXs f'( Ys )d[Y, X] s , { " lo + lo + . hm

'""" .

•

0

"

t+l

"

•+ t

+a

with convergence in ucp. In particular if a = 1/2 then the limit is the F-S integral J�+ f( Ys ) o dXs . Proof We begin by observing that

+

n 1 r 1 ){ ( (d + (Y )) f(Y ) } (X Y J.-t L >. f r,n >. Yr:+1 - rr rr .

•

0

'+1

n

- Xr' ) .

The first sum on the right side of the above equation tends to J� + f(Ys - )dX8 in uc:p. Using the Fundamental Theorem of Calculus, the second sum on the right above equals

n 1 1 1 T 1 s ( ) L J.-t(d>.) ds>.f (Yrr + >. Yr,nf-l - Yr,n ) ) (Yr,+1 - Yrr (X .

•

0

I

0

'+ 1

n T - X ' ),

which in turn equals

where

+

F. s(Yr,+ 1 - Yr,n )) - f'(Yr,n ) I } . •

Since f' and Y are continuous, Also, on [0, t] ,

F
n n ) < [Y, Y] 1/2 [X, X] 1/2 , r r '""" -X ' I t lim sup L I(Yr,+ 1 - Yr,n ) (X t . n --+oo '+1

•

and the result follows by Theorem 30 of Chap. II.

0

Corollary. Let X be a semimartingale, Y be a continuous semimartingale, and f E C 1 . Let an = {Tt }o < i < kn be a sequence of random partitions tending to the identity. Then

288

V Stochastic Differential Equations

with convergence in ucp. Let J.L(d>.) = IO { l / 2 } (d>.) , point mass at 1/2. Then a = J01 AJ.L( d>.) = and we need only apply Theorem 29.

Proof

1/2,

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.

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 = J AJ.L(d>.) . Further suppose that [X, Y] t = J� J8ds; 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 C 1 . Then l t i+ l - xt i ) r lim J.L X f( d>.)( ) ( t >.(t ) yt i+ i l + i n--+00 lo " " { { = f(Ys)dXs + a f' ( Ys)d[Y, X]s , lo + lo+ with convergence in ucp. In particular if a = 1/2 then the limit is the F-S integral J�+ f (Ys) o dXs . Proof We begin with a real analysis result. We let t; denote ti + >.( ti+ l t i ), where the t i are understood to be in an . Suppose a is continuous on [0, t]. Theorem 30.

L •

.

-

Then

which tends to 0. Therefore

t ' l�� 2:: 1 a(s)ds = ).. 1 a(s)ds. n ' tA

.

Moreover since continuous functions are dense in result ( * ) holds for all a in L 1 ( [0, t] , ds ) .

L 1 ([0, t], ds ) ,

the limiting

5 Fisk-Stratonovich Integrals and Differential Equations

Next suppose

289

H is a continuous, adapted process, and set H>. ,n = L Ht , 1 ( t , , t7 J · '

Then taking limits in ucp we have

and using integration by parts, this equals lim { H>.,n

n

· (XY) - ( H>. ,n X_ ) · Y - (H>.,n y ) · X }.

By Theorem 2 1 of Chap. II, this equals

lim{LHt, (xt 7 yt 7 -xt, yt, ) -LHt,Xt, (Yt7 -yt, ) -LHt,Yt, (xt7 -Xt ' ) } n i i i [X, Y] t = J; J.ds, by the result ( * ) we conclude lim L Ht , (x t7 - x t ' )(Y t7 - yt' ) = lim l: Ht, { [X, Yf7 - [X, Y ] t' } ( ** ) n i n i t>. li� J.ds Ht, t = .A lo H8d [X, Y] 8•

However since

=

2:: 1' '

'

We now turn our attention to the statement of the theorem. Using the Mean Value Theorem we have

1 2:: 1 f (�7 )J-t(d.A)(Xt'+ l - xt' ) '

= L f( Yt J(Xt'+ l t

- x t' ) + L t

(* * *)

fol J-t(d.A)f'(YtJ(�; - YtJ(Xt'+ l - xt, )

{ � (d..\) f � s{ f' (Yt, +s (�>. - yt. ) ) - f' (Yt, )}(�>. -Yt. )(X t '+ l_x t , ). + ""' ' � lo lo '

'

The first sum tends in ucp to J f (Y8 )dX8 by Theorem 21 of Chap. II. The second sum on the right side of ( * * * ) can be written as

V

290

Stochastic Differential Equations

:L •

1 M(d>.)f'(YtJ(yt� - YtJ (Xt� - xt• ) 1 + L fo M (d>.) J' (YtJ (�; - YtJ (xti+l - xt; ) . 1

•

The first sum above converges to

by

(

** ) ,

and the second sum can be written as

K>.,n . X

where

'

K;,n Li f'(YtJ(yt� - YtJ l(t;,t,+ 1J (s ) . =

K>., n

Then limn · 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)

Then

(

=

1{ (d>.)F (w)iyt.\ Xt'+ l - ti X l M n ' - Yt, I I

lim '"""' � Jo

n

•

+ s (yt.\' - YtJ ) - f'(YtJ I . IJ'(Yt, t;E
< li�

=0

1 Jl (d>.) Fn {L(yt� - yt, ) 2 } 1/2 {L(Xti+l - Xt, )2 } 1/2 1

•

•

= 0 a.s. and the summations stay bounded in probability. This since lim completes the proof. Since Y is continuous, [Y, X ] = [Y, X] c , whence if a = 1/2 we obtain the Fisk-Stratonovich integral. 0

Fn

Corollary. Let X be a semimartingale, Y a continuous semimartingale, and f be C1. Let [X, Y] be absolutely continuous and let = { tf} be a sequence of non-random partitions tending to the identity. Then

an

with convergence in

ucp.

6 The Markov Nature of Solutions

Proof

Let J.L ( d>. ) = c { 1; 2} (d>.), point mass at apply Theorem 30.

291

1/2. Then J AJ.L(d>.) = 1/2, and 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 C 1 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 t t Xt = Xo + f (s, Xs)dBs + g(s, Xs)ds,

1

1

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. If we 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. I. Assume as given a filtered probability space (!1, :F, (:Ft ) t >O, P) satisfying the usual hypotheses.6

Definition. A process Z with values in JR.d and adapted to IF = (:Ft )t > 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,

E{ f (Zu) [ :Ft } = E{ f (Zu ) [ a{ Zt} } .

One thinks of this a.S "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 (Z. , J ) = E {f (Zt ) IF. } .

Note that if f ( ) = 1 A (x ) , the indicator function of a set A, then the preceding equality reduces to

x

P(Zt E A I Fs) = Ps,t( Zs, l A) · with A, we often write Ps, t(Z8, A) on the right side above.

Identifying 1A When we speak of a Markov process without specifying the filtration of a­ algebras (.Ft)t >o, we mean implicitly that J1 = a {Z8; 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 (Pt)t > O · In the time homogeneous case, the Markov property becomes

P(Zt + s E AI.Ft) = 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 < oo ) = 1 , s > 0,

P( Zr+ s E AI.Fr) = P. ( Zr , A)

or equivalently

E{f(Zr+s)IFr} = P, (Zr , 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 1R.d-valued simple Markov process, then it is easy to see that the process Zt = (Xt , t) is an JR_d+ 1 -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. I. 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

Definition. A function f JR.+ x JR.n exists a finite constant k such that :

293

JR. is said to be Lipschitz if there

---->

i f(t, x) - f (t, y)i < kix - yj, each t E JR.+, and t �-----' f(t, x ) is right continuous with left limits, each x E JR.n . f is said to be autonomous if f(t, x) = f(x). (i) (ii)

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

f! = JR.n

-=
X

f!

/3 ® :Ft p = p where 13 denotes the Borel sets of JR.n , and denotes the Dirac point mass measure at y . For a point w = (y, w) E n, we further define when w = (y, w). X (w) = y, :Ft

r:Y

=

E:y X

E:y

o

Finally let :Ft = nu >t :F�. A random variable Z defined on f! 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 S ect 3 of Chap. IV. Theorem 31. Let zi be semimartingales ( 1 < j < d), HX a vector of adapted processes in ][)) for each x E JR.n , and suppose ( x, t, w) �-----' Hf (w) is /3 ® /3+ ® :F .

measurable. 7 Let FJ be functional Lipschitz and for each x E JR.n , xx is the unique solution of

There exists a version of xx such that (x, t, w) �-----' Xf (w) is /3 ® /3+ ® :F measurable, and for each x, Xf is a cadlag solution of the equation. Proof Let X0 (x, t, w) = Hf (w) and define inductively

1

t

x n +l (x, t,w) i = Htx + L FJ (xn (x, · , ·))s-dZ1. j=l 0 The integrands above are in L, hence by Theorem 63 in Chap. IV there exists d

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 0 of Theorem 62 of Chap. IV yields the result. 7 B denotes the Borel sets on JR.n ; B+ the Borel sets on JR.+ .

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. cesses, Z0 = 0, Xo be as in ( * )

Let Z = ( Z 1 , . . . , zd ) be a vector of independent Levy pro­ and let (fJ ) 1 < j < d, 1 < i < n, be Lipschitz functions. Let and let X be the solution of d

l

x; = Xo + L fj (s�, Xs - )dZ{ j= l 0 Then X is a Markov process, under each pY and X is strong Markov if the fj are autonomous. Proof We treat only the case n = 1. Let T be an lF stopping time, T < oo a.s. Define gr = a{ Z?+u � Z?; u > 0, 1 < j < d } . Then gr is independent of :Fr under P , since the ZJ are Levy processes, as a consequence of Theorem 32 of Chap. I. Choose a stopping time T < oo a.s. and let it be fixed. For u > 0 -y

t

0

define inductively

Y0 ( x, T, u ) = x, d T+u +l yn (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 + 2:. }7 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 g r 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(X0, 0, T + u) = X(X(Xo, 0, T) , T, u).

l

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 :Fr and gr (as well as of :Fr and g r ), we have for any bounded, Borel function h

Ex {h(X(Xo, 0, T + u)) i:Fr } = E{h(X(x, 0, T + u))i:Fr } 1 IR = E{h(X(X (x, O, T), T, u))}1 IR = j (X(x, 0, T)) 1 IR ,

6 The Markov Nature of Solutions

295

where j(y) = E{h(X(y, T, u)} . The last equality follows from the elementary fact that E{F(H, ·)IH} = f (H) , where f(h) = E{F(h, ·)}, if F is independent of 1i and H is 1i measurable. This completes the proof, since the fact that Ex {h ( X ( Xo , 0, T + u))I.Fr } is a function only of X(x, 0, T) implies that Ex {h (X ( Xo , 0, T + u))I.Fr}

= Ex {h ( X ( Xo, 0, T + u))IX(Xo, 0, T)}.

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 = (Z 1 , . . . , zd ) be a vector of independent Levy pro­ cesses, Zo = 0. Then [Zi , ZJ ] c = 0 if i 1:- j, and [Zi , Zi ]r = at, where a = E{ [Zi , zi ]i} . 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{Zf} is an martingale for each i, with E{Zf} = tE{zn. By independence Mi MJ is also a martingale and hence [Mi, M1 ] = 0 by Corol­ lary 2 of Theorem 27 of Chap. II. Therefore [Zi , z1]r = [M i , M1 Jr = 0 as well. Next consider A� [Zi , zi ]t = [M i , Mi ]t. It is an immediate consequence of approximation by sums (Theorem 22 of Chap. II) that Ai also has inde­ pendent increments. Since

Proof =

L2

A� = [M i , M i lr + :L (�M;) 2 , O<s
increments) , and also

by the same reasoning. Therefore [zi , zi ]r = tE{[zi , zi ]i} by subtraction. If the jumps of Z are not bounded, let Jf = Lo<s 1 } . Then zi zi - Ji and Ji are independent Levy processes, and Ji is a quadratic pure jump semimartingale. It therefore follows that [Zi , z1 ] = [Zi , Z1 ] = 0 0 and that [Zi , z ir = [Z i , Z i ] c , and the theorem is proved. =

Let Z = (Z 1 , . . . , zd ) be a vector of independent Levy pro­ and let (fj ) 1 < j < d, 1 < i < n, be non-random F-S accept­ able functions.8 Let X0 be as in Theorem 32 and X be the solution of

Theorem 34. cesses, Zo = 0, 8

F-S acceptable functions are defined on page 278.

296

V Stochastic Differential Equations d

Xf = X6 + L j=l

t

10 f}(s -, Xs- )

o

dZ{

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 [Z i , z1]f = 0 if i 1:- j and [zi , zi]� = ait , ai > 0. 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., Z0 = 0,

such that Z is a semimartingale. Let f and g be Lipschitz functions. Let X0 be as in Theorem 32 and X be the solution of t Xt = Xo + f(s- , Xs-)dZs + Jt g(s, Xs - )ds. o

1

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. n and z is automatically X

extended to n as explained at the beginning of this section. The probability Py = E:y x P is such that Py (Xo = y) = 1, each y E JR.. As in the proof of Theorem 32 for a fixed stopping time T (T < oo a.s.) and x E JR. define inductively for u > 0

-

-

Y 0 (x, T, u) = x rT+u yn+1 (x, T, u) = x + Jr f(v-, yn (x, T, v-))dZv +

rT+u

g(v - , Yn (x, T, v-))dv,

Jr and let X(x, T, u) denote the unique solution of rT+u rT+u X(x, T, u) = x + J f(v -, X(x, T, v-))dZv + J g(v- , X(x , T, v-))d v. r r

6 The Markov Nature of Solutions

297

Next define gT = a{ZT+ u - ZT ; u > 0} fo r the same stopping time T. As in the proof of Theorem 32 we see that yn (x, T, u) is g T measurable for each n > 1 and that X (x, T, u) is gT measurable as well. Next let h be Borel and bounded. measurable and bounded, and let G E Observe that

gT

E{h ( XT)GIFT} = h(XT)E{G I Fr } = h (XT)E{GIZT} = j (Xr , Zr ) , lR is Borel. Since XT +u = X(XT , T, u) by the uniqueness

where j JR2 of the solution, the theorem now follows from the Monotone Class Theorem (Theorem 8 of Chap. I) . 0 :

---t

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 1-L· Indeed for 1-L a probability measure on (JR, B), define -J-L P on f2 by -J-L P (A) = fn�. P ( A) /-L (dx ) , for A E B 0 F. Then the conclusions of the three theorems are trivially still valid for p�-' , and the distribution of X0 is 1-L · Traditionally the most important Markovian solutions of stochastic dif­ ferential equations are diffusions. Suppose as given a space (f2, 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 l , . . . , Bd ) be a standard Brownian motion on JRd , B 0 = 0, and let (fj) 1 < j < d, 1 < i < n, gi be autonomous Lipschitz x

-

=

functions. Let Xb be as in Theorem 32 and let X be the solution of

Then X is a diffusion. If the coefficients U)h<j
=

d

1

Yoi + L fj (Ys) dB� + r g i (Ys)ds, lo J= l o t

t

0

The definition of a diffusion is not yet standardized. We give a general definition.

298

V Stochastic Differential Equations

then Y is diffusion. a

In equation ( ) of Theorem 36 the coefficients Jj 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 !Bt l cannot be represented ) . Another example is that of 1 Btl 1 13 which is a time as a solution of ( 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 f(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 f 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) = 0. Example. Consider the simple system

Xt =

a

1t dB + 1 t aVsds, t Xo + lo V8ds.

vt = Vo +

s

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 (vt) t >o is a true velocity process for X. Using integration by parts we can verify that

6 The Markov Nature of Solutions is an explicit solution for t

e-"'t vt = Vo + 1 =

V. Indeed t 8 Vs ( - o:e "' )ds + 1 e - "' s dv;, t

t

299

t

Vo - o: 1 v;, e-"'8 ds + 1 e - "'8 o:V, ds + 1 e - "' 8 adBs

Vo + lo e- "' 8 adB8 , t

=

and we are done. Since V0 and the Brownian motion B are independent (by construction) , we see that when V0 has a Gaussian distribution then V is a Gaussian process. If o: is negative and we take a 2 = -1/(2o:), then V is a stationary Gaussian process. Example. Consider next the equation t Xs - ds + Bt, (0 < 1) t < Xt = 1 s 0 -

lo

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 t 1 < t0 then the solution for [0, t0] agrees with the solution for [0, h] on the interval [0, h]. Thus we have a solution on [0, 1). If we can show that limt__, 1 Xt = 0, then the solution extends by continuity to [0, 1] and we will have constructed a diffusion X on [0, 1] with X0 = X1 = 0, 0 for

known as the Brownian bridge. 1 An application of integration by parts shows that the solution of the Brow­ nian bridge equation is given by t 1 f < 1). (0 t dEs Xt = (1 - t) < }0 1 - s

Write Xt = f(t)Mt , where f(t) = (1 -t) and Mt = J�(1 -s) - 1 dB8• To see that limt __, 1 Xt = 0, we study Mt, making the change of variables t = u(1 + u) 1 . Then 0 < t < 1 corresponds to 0 < u < oo , and define -

N is clearly a continuous 9u 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, that 11 Nt = 0 a.s. lim t -->oo

t

-

1 0 The Brownian bridge is also known as tied down Brownian motion, and alterna­ tively as pinned Brownian motion. 1 1 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 1 , w) a.s. We then have, replacing t with (1 - t)

=

0 a.s. Thus, limt_, tN1; t = 0 o

lim ( l - t)M1 / ( 2 -t) = 0 a.s., tlim ---> 1 ( l t)N1 ; ( 1 - t) = t---> 1 and therefore limu _, 1 (1 - u)Mu = 0 a.s., and hence limt_,1 Xt = 0 a.s. We now know that X is a continuous diffusion on [0, 1], and that X0 = X1 = 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 f01 11X sJ ds < oo a.s. To see this calculate E { xn, 0 < t < 1, -

By the Cauchy-Schwarz inequality

Therefore

1 l Xs l 1 E { I Xs l } { { E{ }0 1 - s ds} = }0 1 - s ds 1 Js( 1 - s) ds < oo < { ' - }0 1 - s

whence J01 11Xs� ds < oo a.s. Therefore the solution X is a semimartingale on [0, 1]. Finally we remark that one can construct a similar Brownian bridge from any value a to any value b, on an interval of arbitrary finite length T . Using the interval [0, r] one has the equation

Xt = a + with solution

1t b - Xs ds + Bt, 0

T - S

1

0
t 1 t bt dB8 , a(1 - -) + - + (r - t) 0 T

Xr = b.

T

T - S

T,

0
T,

7 Flows of SDE: Continui ty 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 JRn }Rn for ( t, w) fixed, or as mapping JRn vn , where vn denotes the space of cadla,g functions from JR + to }Rn , equipped with the topology of uniform convergence on compacts. It is important to distin­ guish between vn and ]]))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 ( cf., 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 JRn . We begin with continuity. We consider a general system of equations of the form :

___.

___.

,

where Xf and Hf are column vectors in JRn , Z is a column vector of m semimartingales with Z 0 0, and F is an n x m matrix with elements (F�). For fixed, for each y we have that X t = xr -Xt is a solution of the equation

X

=

t

Xt Hf - H{ + 1 F(X)s - dZs, =

where F(Y) = F(Xx + Y) Theorem 37. Let Hx be

- F(Xx). processes in ]]))n , and let x f--+ Hx : JRn -. ]]))n be prelocally Lipschitz continuous in SP, some p > n. Let F be an n m matrix of functional Lipschitz operators (F�), 1 < i < n, 1 < a < m. 12 Then there exists a function X (t, w, X ) on JR+ n }Rn such that (i) for each x the process Xf(w) X(t, w, x) is a solution of (*), and (ii) for almost all w ' the flow X f--+ X ( . , w, X ) from }Rn into vn is continuous in the topology of uniform convergence on compacts. x

X

X

=

12

See page 250 for the definition of functional Lipschitz.

302

V Stochastic Differential Equations

Proof.

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 which is finite a.s. Then by conditioning 1 3 we can assume without loss of generality that this Lipschitz constant is non-random, and we call it c < oo. By replac­ and then by replacing with given by ing with + J� we can further assume without loss of generality that = = Then for {3 = Cp(c) , by Theorem 5 we can find an arbitrarily large stopping time T such that zr - E S({3), and is Lipschitz continuous in on T) . Then by Lemma (preceding Theorem 7) we have that for the solution of ( )

K

Hf Xf F (0)8_dZ8 , F G F(O)t, G(Y)t F(Y)t F(O) 0. x H 2 SP [0 , X I XT- I -sp < Cp(c, Z)! ! (Hx - HY f- l -sP, for any > 2, and some (finite) constant Cp(c, Z). Choose > n, and we have E{sup I X: - Xfi P } < Cp(c , Z)K ! x - y i ! P , s< T due to the Lipschitz hypothesis on x Hx. By Kolmogorov's Lemma (The­ orem 72 of Chap. IV) we have the result on JRn x [0, T) . However since T was arbitrarily large, the result holds as well on JR n f2 JR+. **

p

p

f-+

X

0

X

For the remainder of this section we will consider less general equations. Indeed, the following will be our basic hypotheses, which we will supplement as needed.

zo: are given semimartingales with Z0 = 0, 1 < a < Hypothesis (H2) . ]Rn lR are given functions, 1 < i < n, 1 < a <

Hypothesis (Hl ) .

f(x)

f� denotes the n x m matrix :

--+

(f�(x)).

and We will study the system of equations

Xi =

m Xi + L rn o: =l J0

f�(Xs_)dZ':,

m. m,

(0)

which we also write

(00) and are column vectors in JRn , where it is understood that is an n x m matrix, and is a column vector of m semimartingales. To study the differentiability of the flow we will need a more general theo­ rem on existence and uniqueness of solutions. Indeed, we wish to replace our customary (global) Lipschitz condition with a local Lipschitz condition.

Z

1 3 See the proofs

Xt x

of Theorems 7, 8, or 15 for this argument.

f(X8_)

7 Flows of SDE: Continuity and Differentiability

303

Definition. A function f JR.n � JR. is said to be locally Lipschitz if there exists an increasing sequence of open sets JR.n and f is such that Lipschitz with a constant on each :

Kk

A

Uk Ak =

k A k.

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) .

,

Let Z be as in (Hl) and let the functions (!�) in (H2) be locally Lipschitz. Then there exists a function ((x, w) : JR.n x n [0, oo] such that for each x ( ( x, ) is a stopping time, and there exists a unique solution of

Theorem 38.

�

·

(00) ) with lim supt_, <;( x, . ) IIXt ll = oo a.s. on {( < oo } . Moreover ((x, semi-continuous, strictly positive, and the flow of X ((x,w) isonlower xis continuous [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 ·

up to �--t

·

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 ll = + oo on < oo } . Note however that if the coefficients (!� ) in (H2) are (globally) Lipschitz, then a.s. oo for all (Theorem 7).

{T

At

(=

x

( t)

be open sets increasing to JR.n such that there exist h , a Let sequence of c= functions with compact support mapping JR.n to [ 0, 1] such that h denote the solution (continuous 1 on For each f. we let X t in of the equation t g Xs )dZs Xt

Proof

t=

At.

t( w, x) x) = X + lo t( 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, Be) we have that Xt = Xt(00)+ l . by the uniqueness of the solutions, since they both satisfy equation We wish to show that the relation Xt( , w , x) = Xt + l ( · ,w,x) on [ O ,St(w,x)) (L) holds for all x E JR.n simultaneously. Choose an x and let ,

-

·

304

V Stochastic Differential Equations

Ax = {w : Xt(· , w, x) = Xt+l ( · , w, x) on [O, St(w, x))}. Then P(Ax ) = 1. We set A = Ux E IQin Ax , where Qn denotes the rationals in JRn . Then P(A) = 1 as well, and without loss of generality we take A = 0. n Next for arbitrary E n let with Y y

lR ,

Yn

-+

y,

n E Q . Then

St (W , Yn ) , St(w, y) < lim inf oo n -> and therefore the relation ( L) holds for y as well, by the previously established continuity. Observe that St (w , x) < S£+ 1 (w , x), and let ( (x, w) = supt St (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 Xt (·, w, x) on [O, St (w, x) ) , each C > 1 . Indeed, on [O , St(w, x)) we have X(-, w, x) = Xt (·, w, x) , and since XtC , w, x) E At , we have ht(Xt(·, w, x ) ) = 1 , wherefore Xt is a solution of (00) on [0, St) · 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 Xt 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 Aj by right continuity and the definition of St; therefore X(s, w, x) or X(s-, w, x) is in Aj. This shows that lim supt_, ( IIXt ll oo 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, ·) oo 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 B 1 and B 2 be two independent Brownian motions on lR and let Bt - Bf + iB'f , where i2 - 1 . Consider the stochastic differential equation

=

=

=

=

=

=

=

=

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) - 1 and Zt = f(Bt), then f ' (Bt) - Z'f , and since Bt = Bf iB'f , we have by Ito's formula

=

+

=

7 Flows of SDE: Continuity and

Differentiability

305

For z fixed we know that P(3 t : Bt - 1 I z ) 0, since Brownian motion in JR2 a.s. does not hit a specified point. Therefore Z does not have explosions in a finite time for each fixed initial condition Z0 = z. On the other hand for any wo and t0 fixed we have Bt0 ( Wo ) zo for some zo E C, and thus for the initial value z = 1 I zo we have an explosion at the chosen finite time t0 . Thus each trajectory has initial conditions such that it will explode at a finite time, and P{w : 3 z with ((w, z) < oo} 1 . 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 (!� ) are at least C1 )

=

=

=

-

=

(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 t x: = Xi + lo f�(Xs - )dZ'; (D) t &fi Dkti = ukJ:i + 10 aXa:j (Xs - ) Dk1 s - dzo:s . 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 (!�) are locally Lipschitz, then so also are the coefficients of (D). Theorem 39. Let Z be as in (Hl) and let the functions (!�) 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 (!�) are globally Lipschitz then ( = oo. Let Dk (t, w, x) 8�k X(t, w, x). Then for each process (X(·,w, x),D(·, w, x)) is identically cadlag, and it is the solution Xoftheequations ( 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 of X i 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 : \fx E B, ((x, w) > N} the function x X ( t, w, x) is continuously differentiable on B. However by Theorem 38 we know that for each w E r, the image of X as x runs through B is compact in vn with 0 < t < N, hence it is contained in a ball of radius R in JRn, for R sufficiently large. We fix the radius R and we denote by K the ball of radius R of JRn centered at 0. Let

Xk

�--t

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(A I A) = 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 JRn lR be c= 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(f3) for a {3 which will be A = {w

:

:

for x

--+

specified later . If we were to proceed to calculate formally the derivative with respect to of Xi , we would get

Xk

Therefore our best candidate for the partial derivative with respect to the solution of the system

Xk is

1 4 These derivatives are also known as derivatives in the generalized function sense.

7 Flows of SDE: Continuity and Differentiability

D

307

and we can make (Di).byNaturally X(s,w,x), Xs x rewriting the above equation

and let be the matrix explicit this dependence on

=

as

where the summations over now show that =

a

and j are implicit (Einstein convention) . We Fix E !Rn and let is continuous Then

. (D�, . . . , DJ:) i n Dk x, x y Vs(w) Dk(s, w, x) - Dk(s,w, y). V: i 1t { of� x _ of� (Xsy ) JY.Y } dZsa. jx D X ) ( s ks ;::. - ks ;::. - - UXj t UXj 1t { ofUXj ;::. � (XsY- )} Djxks - dzsa. ;::. � (Xsx- ) - ofUXj 1t of� ;::.UXj (XsY- ) { Djxks - - d,ksY- } dZsa. Hl(x, y , k) lot �f�xi (Xf_ )Vj_ dZ� t Hl 1 Vj_dYji, where Yji I::' J; ��: (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 there exists a constant Cp ( Z) such that =

=

0

=

0

+

0

+

=

+

=

1

However we can also estimate IIHII sp . If we let as

Ji

=

� � j= 1

{ Oof� (XsX- ) - of� Y ) } jx OX ( Xs - D ks - ' XJ.

.

J

I::' J� J�8dZ';, and therefore by Emery's inequality (Theorem 3) i i Z I I H= , II H I I sp < cpiiJIIsp which in turn implies then Hf = we have that

1

-

IIVIIsp -

-

< Cp (Z) IIJIIsp - .

We turn our attention to estimating II Jll SP . Consider first the terms of� ( X x ) _ of�

;::.ux · s - uX;::. · (XsY- ). J

J

308

V

Stochastic Differential Equations

1, the functions �;:. are Lipschitz in By the simplifying assumptions of takes its values in Therefore Theorem 37 applies, and as we and saw in its proof (inequality * * * )) we have that

xx

K,

Step

K.

J

( of� (Xsx ) _ of� (XsY ) x Y < < K(p, Z) l x - Y l · X X K I I s 2p uX · ux · s2p Next consider the terms D�� . We have seen that these terms are solutions of the system of equations V,ktx = k 1t Dtksx- of� (Xsx- ) dZsa. uxg �

�

J

J

61 +

�

0

and therefore they can be written as solutions of the exponential system

t DJktx = oki 1 Dksex- d J x with YJsx = J; �;� (X�_ )dZ�. As before, by Lemma 2, +

0

Y.

ls '

J and using the Cauchy-Schwarz inequality gives I J I sp < l ��: (x:_) - ��: (Xf_) l s2p i D�x l s2p < C2p(Z) I x - Yl l ,

Recalling the definition of

A

which in turn combined with previous estimates yields

II V IIsp < C(p, Z) l x - Y l ·

Since V was defined to be that (with >

Vs(w) = Dk(s,w,x) - Dk(s,w, y ), we have shown

p n) E { sup I Dk(s, w, x) - Dk(s, w, y ) I P } < C(p, Z)PI I x - Y I P , s
and therefore by Kolmogorov's Lemma (Theorem 72 of Chap. IV) we have the continuity in of the solution of equations 3. In this step we first show that ( *4 ) (and hence also the solution of the equations of the second line of in the sense of distributions is the partial derivative of in the variable (i.e., generalized functions). Since in 2 we established the continuity of in we can conclude that is the true partial derivative.

Step

Dk x,

x Dk(t,w,x). X

Dk

Dk(t,w,x), n2 Xk Step

(D))

7 Flows of SDE: Continuity and Differentiability

309

Let us first note that with the continuity established in Step 2, by increas­ we can assume further that Dk (s, w, x) E also, for ing the compact ball 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

K,

K

Let Ur 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 J(Ya + )�dZ8 • yt = Y

+ lo

t

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 r such that limq --+oo X(ur. ) = X and limq --+oo Dk (ur. ) = D k , in the sense of distributions, considered as functions of x. We now enlarge our space 0 exactly as in Sect. 6 (immediately preceding Theorem 31 ) q

:

n = IR2n � = B2n

F

X

n

®Ft, Ft = n F� , u> t

where B2n denotes the Borel sets of IR2n . Let .>. be normalized Lebesgue mea­ sure of Finally define P = >. x P. We can as in the proof of Step 2 assume that Z E S((3) for (3 small enough and then, S2 , --+oo (X(ur), D(ur) = (X, D) in rlim by Theorem 16. Therefore there exists a subsequence rq such that

K.

M = L: sup jj (X(rq ) , D(rq ) ) - (X, D) I I q=l t 00

E L 1 ( dP).

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

= (X, D) (X(rq), D(rq)) qlim oo ---+

>. a. e. Further, it is bounded by the function M(w, · )+I I (X (t, w, · ) , D( t, w, · )) I

which is integrable by hypothesis. This gives convergence in the distributional sense, and the proof is complete. D

39 as a theorem. Theorem 40. Let Z be as in {H1) and let the functions (!�) in {H2) have locally Lipschitz derivatives up to order N, for some N, 0 < N < oo . Then there exists a solution X (t, w, x) to We state the following corollary to Theorem

Xf = Xi +

rt

f�(Xs - )dZ';, 1 < i < n, L o a = 1 J° m

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 ( = oo . 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 c N - 1 , whence X E CN. 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 cN + l (JRn), for each i and a , then (!�) 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

xj

= Xi + L= 1 lot° F� (X)s- dZ';, m

1 < i < n,

a semimartingales Z"' are assumed to have continuous paths

with where the Zo = 0. 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 1Rn-valued function

8 Flows as Diffeomorphisms: The Continuous

Definition. The flow

Case 311

( ) is said to be weakly injective if for *

P{w : 3 t : X(t, w, x) = X(t,w, y)} = 0. Definition. The flow

3

For convenience we recall here a definition from Sect. of this chapter. Definition. An operator F from Jl))n into lill is said to be process Lipschitz if for any X, Y E Jl))n the following two conditions are satisfied. (i) For any stopping time T, xr- = yr- implies F (X) T = F(Y) T- . (ii) There exists an adapted process such that

KEL

-

Actually, process Lipschitz is only slightly more general than random Lip­ schitz. The norm symbols in the above definition denote Euclidean norm, and not sup norm. Note that if F is process Lipschitz then F is also functional Lipschitz and all the theorems we have proven for functional Lipschitz coeffi­ cients hold as well for process Lipschitz coefficients. If f is a function which is Lipschitz (as defined at the beginning of Sect. then f induces a process Lipschitz operator. Finally, observe that by Theorem we know that the flow of equation ( ) is continuous from !Rn into !Rn or from !Rn into vn a.s., where vn has the topology of uniform convergence on compacts.

3) 37

*

Let za. be continuous semimartingales, 1 < a < m, H a vector of adapted cadlag processes, and F an n x m matrix of process Lipschitz operators. Then the flow of the solution of

Theorem 41.

Xt = X + Ht + lo

t F(X)

8_ dZ8

is weakly injective. 1 5 Proof. Let x, y E !Rn , x =f- y. Let xx, XY denote the solutions of the above equation with initial conditions x, y respectively. We let u = x - y and U = xx - XY. We must show P{w : 3t : Ut (w ) = 0} = 0. Set V = F(Xx ) _ F(XY ) _ . Then V E L and J V I < K J U J . Further, the processes U and V are

related by

=

t

Ut =

Ut = u + lo V.dZ8•

=

0}; the aim is to show P(T oo ) = 1. Since is Let T inf {t > 0 : continuous the stopping time T is the limit of a sequence of increasing stopping 15

We are using the Einstein convention on sums.

U

V

312

Stochastic Differential Equations

1 [o ,sk ]

1 [o ,T)

times Sk strictly less than T. Therefore the process is = limk -+oo predictable. We use Ito's formula (Theorem of Chap. II) on [0, T) for the function f(x) = log Note that

l xl l ·

32

af OXi ()2 f ax2z af

1 8ll x l -

1 -

Xi

l x l axi l x l l x l 1 2xi al l x l l x l 2 l x l 3 OXi 2x· x · · OXiOXj l xl 4 t

t

J

Xi .

l xl 2 ' 1 2x2 l xl 2 l xl 4 ' t

•

Therefore on [0, T),

log I Ut II - log II u II t id i - "' f L..-Jo 11 Us ii 2 Us Us +

!

1

I

t 1 t i . 1 L.."' f d [Ui , Uil s - "' f us u; d[Ui , Uj l s · �Jo I I Us l l 4 2 Jo IIUs ll2 I

I,J

Since dUi = E a V i •"' dZ"' , the foregoing equals

All the integrands on the right side are predictable and since II VII < KIIUII they are moreover bounded by K and K2 in absolute value. However on tends to - oo as t {T < oo} the left side of the equation, log IIUt ll - log increases to T; the right side is a well-defined non-exploding semimartingale 0 on all of [0, oo). Therefore P(T < oo) = 0, and the proof is complete.

!l ul ,

In the study of strong injectivity the stochastic exponential of a semi­ martingale (introduced in Theorem of Chap. II) plays an important role. Recall that if Z is a continuous semimartingale, then Xo£(Z) denotes t he (unique) solution of the equation t Xt = Xo + XsdZs ,

37

lo

= exp{ Zt - � [Z, Z] t } . In particular, P(infs < t £(Z) s > 0) = 1. Theorem 42. For x E !Rn , 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 (Tt)t> l increasing to oo a.s. such that I (Hx - HY ) Tt l sr < K l x - Yl l , each and £(Z)t

8 Flows as Diffeomorphisms: The Continuous Case

for a constant K and for some r semimartingales. Then the functions

£ > 1,

>

n. Let Z

=

313

(Z1 , . . , zk ) be k .

t X I-+ lo H: dZ8 X 1--+ [Hx Z, Hx Z] t ·

·

have versions which are continuous as functions from !Rn into V, with V having the topology of uniform convergence on compacts. Proof. rBy Theorem 5 there exists an arbitrarily large stopping time T such that z - E H00 • Thus without loss of generality we can assume that Z E H00 , and that Hx is bounded by some constant K, uniformly in x. Further we assume I Hx - HY l sr < Kl l x Y l · Then -

-

-

t H:_dZs - t Hf_ dZs l n < CE{sup I Hf_ - Hf- lni i Z I -H= I t lo t lo < K l x - Yl n! Z I H = ,

E{sup

where we have used Emery's inequality (Theorem 3). The result for J H;_ dZ8 now follows from Kolmogorov's Lemma (Theorem 72 of Chap. IV) . For the second result we have

I [Hx Z, Hx Z] - [HY Z,HY Z] l sr = I J {(H:_)2 - (Hf_ ) 2}d[ Z, Z] sl l sr k = I i,Lj=l j(H;i ) + Ht (H:J - Hfj )d [zi , zj ]sl l sr, < 2K I Z I koo I HX - HY I sr , ·

·

·

·

-

and the result follows.

0

Let F be a matrix of process Lipschitz operators and xx the with initial condition x, for continuous semimartingales za. , Fix x, y E !Rn . For r E IR there exist for every x, y E !Rn with 1 < a < x i y (uniformly) locally bounded predictable processes H0 ( x , y), J a. .f3 ( x , y), which depend on r, such that Theorem 43. solution of ( *) m.

where

314

V Stochastic Differential Equations

x, y

U

XY,

F(X x )_ - F(XY)_.

Ito 's Fix E !Rn and let = xx V= formula applies since is never zero by weak injectivity (Theorem 36). Using the Einstein convention,

Proof.

U

I U I r = l x - Y l r + J ri 1 Usl l r - 2 u;du; + � j r{(r - 2) J Usl l r-4U;u1 + o) J I Us l r -2}d[Ui , UJ]s·

Let ( ) denote Euclidean inner product on !Rn . It suffices to take ·

,

·

(where V"' is the a-th column of V); and

One checks that these choices work by observing that Finally the above allows us to conclude that t JJ Ut JJ r =

dUf = E:' �i ·"'dZf. 1

J x - yJ J r + fo J UsJJ rdAr (X, Y)s,

and the result follows.

D

Before giving a key corollary to Theorem 43, we need a lemma. Let H= = 0 such that be the space of continuous semimartingales with has a (unique) decomposition X= N+A where is a continuous local martingale, is a continuous process of finite variation, = = 0, and such that and f0= I are in L= . Further, let us define

X

A [N, N]=

N No Ao

Lemma. For every p, a < < a , then if

oo ,

Xo

X

JdAs

there exists a constant C(p, a) < oo such that < C(p, a ) .

J £ (X) J sp Proof Let X = N + A be the (unique) decomposition of X. Then 1 E{ s p (X) I £ Jl�p � exp{p(Xt [X, X] t) }} (recall that X * = sup I Xt l ) t < E{exp{pN * + pa}} = ePa E{ epN * }, J X J I .Hoo

=

-

2

8 Flows as Diffeomorphisms: The Continuous Case

315

since I Atl < 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 , N] t , where B is a Brownian motion defined with a different filtration. Therefore since [N, N] oo < a 2 , we have and hence E{ eP N * } < E{ exp {pB� } }. Using the reflection principle ( Theo­ rem 33 of Chap. I) we have 2

0

Note that in the course of the proof of the lemma we obtained 21/P exp{ a + pa 2 /2}.

C(p, a) =

Corollary. Let - oo < r < oo and p < oo , and let (Ar (x, Y)t)t>O be as given in Theorem 43. Then £ (Ar (x, y)) is locally in SP , uniformly in x, y.

Proof.

We need to show that there exists a sequence of stopping times T;_ increasing to oo a.s., and constants C;_ < oo, such that II£( Ar (x, y) ft llsp < C;_ for all x, y E �n , x =J y. - oo By stopping, we may assume that za and [za , zi3] are in and that I a l and I Ja ,/3 1 < b for all (x, y),-1oo< a, (3 < m. Therefore II Ar (x, y)II.Hoo < C for a constant C, since if X E and if K is bounded, predictable, then and IlK · X II .Hoo < II K IIs oo II X J I.Hoo , as can be proved exactly K·X E analogously to Emery's inequalities (Theorem 3). The result now follows by the preceding lemma. 0 -

H

H

H00

H

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.

Let za be continuous semimartingales, 1 < a < and F an matrix of process Lipschitz operators. Then the flow of the solution of

Theorem 44.

n

x m

m,

is strongly injective on �n . Proof. It suffices to show that for any compact set C C �n , for each N, there exists an event of probability zero outside of which for every x, y E C with X =J y, inf II X(s, w, x) - X(s, w, y)ll > 0. s
316

V

Stochastic Differential Equations

x, y, s have rational coordinates. By Theorem 43 a.s. I X (s, w ,x) - X( s , w , y)l i r = J x - y1 J r£(Ar (x, y))s . The left side of the equation is continuous (Theorem 37) . As for the right side, will be continuous if we can show that the processes H'; ( x, y ) and £J<;( A,f3r(x,(x, y),)) given in Theorem 43, verify the hypotheses of Theorem 42. To y this end, let B be any relatively compact subset of !Rn x !Rn \ {(x , x)} (e.g. , n where are open balls in R with disjoint closures) . Then B B = B1rxisBbounded B , 1 2 2 B for any real=number r. Without loss we take r = 1 x - yJLetJ (x, ) = xonx -XY, Jhere. V(x, y) F (X x )_ - F (XY)_ , and let V"'(x,y ) y U be the a-th column of V. Then for ( x , y ) and ( x' , y' ) in B we have ( ** ) H"'(x,y ) - H"'(x' , y') = ( J U(x, y) JI- 2 - I J U(x', y')l l - 2)(U(x, y), V"'(x,y)) + J U(x' , y' )l l - 2 (U(x, y ) - U(x' , y' ), V"'(x, y )) + J U(x', y' ) I - 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 II J U (x, y) J l - I U(x' , y' )l l l( I U (x, y) J l + J U (x' , y' ) I J I ) J U (x' y) J I IJ V"'(x' y) J I J U(x, y ) J 2 J U (x' , y' ) J 2 < K I J U(x, y) - U(x', y' ) I J ( I J U(x, y ) J + I J U (x' , y' )l i ) J U(x' , y' ) J - 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( J U(x,x') l + I U(y, y' ) J )( J U(x, y ) J + J U(x' , y') J ) J U(x' , y' ) J I - 2 = K( J x - x' J E (AI(x, x' )) + J y - y'J J E (A t (y,y' ))) · ( I J x - yi J £(A I (x, y)) + I J x' - y'I J £(AI (x' , y'))) J l x' - y'I J - 2£(A- 2 (x' , y')) < K1 I J (x, y ) - (x', y' )I I (£(A 1 (x, x')) + £(A . (y ,y')))(£(AI(x, y )) + £(A 1 (x' , y' )) )£(A - 2 (x', y' ) ). By the lemma following Theorem 43, and by Holder's and Minkowski's inequalities we may, for any < oo, find stopping times Tt increasing to oo a.s. such that the last term above is dominated in SP norm by Kt i J (x, y )- (x' , y') J for a constant Kg corresponding to Tp_. 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 same is true for J<> , f3 , and therefore Theorem 42 shows that (x, yand) E [B. The are in (x, y ) on B. (Actually we are using a local , Ar ] 42continuous Aversion r ofArTheorem with (x, y ) E B IR2n instead of all of IR2n ; this is not a problem since Theorem 42 extends to the case x E W open in !Rn , because Let

p

C

317

8 Flows as Diffeomorphisms: The Continuous Case

Kolmogorov's Lemma does-recall that continuity is a local property.) Finally are continuous in since and E B we deduce that is continuous in E We have shown that both sides of

, Ar] JR n : (x, y) (x, y)) £( Ar [Ar{(x,y) Ar 2 x i= y}. IIX(s , w, x) - X(s, w, Y ) l r = l x - Y l r £(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.

D

Let za be continuous semimartingales, 1 < a < and let F be an n matrix of process Lipschitz operators. Let X be the solution of ( *) . Then for each N < oo and almost all w inf I X (s,w,x ) l = oo. lim ->oo l x ll s< N Proof By Theorem 43 the equality Theorem 45 . x m

m,

is valid for all r E JR. 0 let yx = For Theorem 41.) Then

I xx - X0 1 - 1 . (Note that yx is well-defined by ( ) I Yx l = l x l - l £( A - I(x ,O) ) I Yx - YYI < IIX x - XYIIIIXx - X0 I -1 I X Y - X0 l -1 = l x - Yl l l x i -1 I Y I - 1 £( A 1 (x, y) )£( A - I ( x, 0) )£( A - I (y, 0) ) . Define y oo = 0. The mapping x � x l x l -2 inspires a distance d on JRn \ { 0} b d(x, Y ) = lllxi-i i iyllI I " I ndee d ' x Y ( )2 x X Yl l l Y l l x l 2 - I I Y I 2 II 2 = l x i i Y I x i=

***

Y

By Holder ' s inequality we have that

I £(A 1 (x , y) )£(A - l (x , O)£(A-l (y, 0)) llsr < 1 £ ( A l (x, y)) I -s3r 1 £ ( A-l (x, o)) ll -s3r 1 £ (A - l (y, o)) I -s3r

and therefore by the corollary to Theorem 43 we can find a sequence of stop­ ping times (Te)e>I increasing to oo a.s. such that there exist constants Ce with ( using * * *))

(

Next set

318

V

Stochastic Differential Equations A

yx =

{ yyxoollx ll-02

'

= '

0 < llxll < oo , llxll = 0.

Then II (Y x - YY ) Tt llsr < Cf llx - Y llr on JR.n , and by Kolmogorov's Lemma (Theorem 72 of Chap. IV) , there exists aA jointly continuous version of (t, x) � �x , on JR.n . Therefore lim l x ii ->O yx exists and equals 0. Since 0 (Yx ) -1 = II X x ll x ll -2 - X0 ll, we have the result.

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 : JR.n ....-. ]Rn be the flow cp(x) = X(t, w, x). Then for almost all w one has that for all t the function cp is surjective and moreover it is a homeomorphism from lR. n to lR.n . Proof. As noted preceding Theorem 41, the flow cp is continuous from JR.n to vn , topologized by uniform convergence on compacts; hence for a.a. w it is continuous from lR.n to lR.n for all t. The flow cp is injective a.s. for all t by Theorem 44. Next observe that the image of lR.n under cp, denoted cp(lR.n ) , is closed. Indeed, let cp(JR n ) denote its closure and let y E cp (JRn ) . Let (x k ) denote a sequence such that limk -> oo cp (xk ) = y . By Theorem 45, lim supk ->oo ll x k I < oo, and hence the sequence (x k ) has a limit point x E JR. n . Continuity implies cp( x) = y , and we conclude that cp(lR.n ) = cp(JRn ) ; that is, cp(lR.n ) is closed. Theorem 46.

.

Then, as we have seen, the set {x k } 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- 1 is continuous. Since cp is a homeomorphism from lR.n to cp (lR.n ), the subspace cp(JR. n ) of lR.n is homeomorphic to a manifold of dimension n in lR.n ; therefore by the theorem of the invariance of the domain (see, e.g., Greenberg [84, page 82]), the space cp(lR.n ) is open in lR.n . But cp (lR.n ) is also closed and non-empty. There is only one such set in lR. n that is open and closed and non-empty and it is 0 the entire space JR.n . We conclude that cp ( lR.n ) = JR.n . Comment. The proof of Theorem 46 can be simplified as follows: extend to the Alexandrov compactification JR� = lR. n U { oo } of lR.n to cp by cp (x) =

{

cp( x ) , oo ,

cp

X E JRn ' X = 00 .

Then cp is continuous on JR� by Theorem 45, and obviously it is still injective. Since JR.� is compact, cp is a homeomorphism of JR.� onto cp(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 cp(JR�) = JR.� .

8 Flows as Diffeomorphisms: The Continuous Case

319

We next t urn our attention to determining when the flow is a diffeomor­ phism of JR.n. Recall that a diffeomorphism of JR.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

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 n matrices is not commutative, it is also possible to consider right stochastic integrals, denoted where X is an

x

where Z is an n x n matrix of semimartingales and H is an n x n matrix of (integrable) predictable processes. If denotes matrix transpose, then 1

(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, Z] denote n x n matrix-valued processes here.

Let Y, Z be given n x n matrices of semimartingales, H an n x n matrix of locally bounded predictable processes. Then, (i) Y"tZt - YoZo = f� Ys _ dZs + f� (dYs)Zs - + [Y, Z]t i (ii) [H · Y, ZJ = H · [Y, ZJ; and (iii) [Y, Z : HJ = [Y, ZJ : H. Moreover ifF is an n n matrix of functional Lipschitz operators, then there exists a unique n n matrix of ]])) -valued processes which is the solution of

Theorem 47.

x

x

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 D of Theorem 7. Theorem 4 7 allows the definition of the right stochastic exponential.

V

320

Stochastic Differential Equations

Definition. The right stochastic exponential of an n x n matrix of semi­ is the ( unique ) matrix-valued solution of the denoted martingales equation

t: R (Z), t Xt = I + lo (dZs)Xs - ·

Z,

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 Theorem 48. Let be an n x n matrix of continuous semimartingales with

t:R (Z) E(Z ') ' . Z R and t: ( -Z + [Z, Z]) are inverses; that is, E(Z)t: R (-Z + Zo 0.=Then E(Z) [Z, Z]) I. Proof Let U = E(Z) and V = t:R (- Z + [Z,Z]). Since U0V0 = I, it suffices to show that d (Ut vt) = 0, all t > 0. Note that dV = ( -dZ + d[Z, Z])V, (dU)V (UdZ)V, and d[U, V] Ud[Z, V] = -Ud[Z,Z]V. =

=

=

Using Theorem 47 and the preceding,

d(UV) UdV + (dU)V + d[U, V] U( -dZ + d[Z, Z]) V + UdZV - Ud[Z, Z]V =

=

=

0,

0

and we are done.

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 be continuous semimartingales with = 0,

1 , , zm ) zg (Z 1 < i < m, and let (!�), 1 < i < n, 1 < a < m, be functions mapping to with locally Lipschitz partial derivatives up to order N, 1 < N < and bounded first derivatives. Then there exists a solution X ( t, w, x) to •

•

•

JRn

oo,

JR,

such that its flow r.p : x ---> X(x, t, w ) is N times continuously differentiable on JRn . Moreover the first partial derivatives satisfy the linear equation ( l < i < n). where 8f. is Kronecker's delta.

9 General Stochast ic 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 = oo is included in the statement. The explicit equation for the partial derivatives comes from Theorem 39. Let denote the n x n matrix-valued process

D

( 0)

The process by

D is the right stochastic exponential [R (Y), where Y is defined f � j � i dzsa . dYs = (X 8 ) a= l a

L..J ax J

,

.

Combining Theorems 48 and 49 and the above observation we have the im­ portant following result . Theorem 50. With the hypotheses and notation of Theorem 49, the matrix a.s. is non-singular for all t > 0 and x E Theorem 5 1 . Let be continuous semimartingales and let (!�),

Dt

(Z 1 , . . . , zm )

JRn ,

be functions mapping JR n to JR, with partial derivatives of all orders, and bounded first partials. Then the flow of the solution of

1 < i < n , 1 < a < m,

t m r x: = Xi + L Jo f�(Xs)dZ';, 1 < i < a= 1 0 is a diffeomorphism from JRn to JRn . Proof. Let r.p denote the flow of X. Since (f�h< i< n, l
D

9 General Stochastic Exponentials and Linear Equations

Z

Z0

Let be a given continuous semimartingale with = 0 and let the unique solution of the stochastic exponential equation

t

E(Z)t denote

Xt 1 + l XsdZ8• Then Xt E(Z)t = exp { Zt �[Z, Z]t} ( cf. , 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

HE

Z

where ]]J) (cadlag and adapted) , and is a continuous semimartingale. A unique solution of **) exists by Theorem 7. It is written '

( £H (Z). Theorem 52. Let H be a semimartingale and let Z be a continuous semi­ martingale with Z0 0. Then the solution £H (Z) of equation (**) is given by t{ £H (Z)t £(Z)t{Ho + lo+ £ (Z) ; 1 d(Hs - [H, Z]8)}. Proof We use the method of "variation of constants." Assume the solution is of the form Xt CtUt , where Ut £(Z)t, the normal stochastic exponential. The process C is cadlag while U is continuous. Using integration by parts, dXt Ct _dUt + UtdCt + d[C, U]t Ct- Ut dZt + UtdCt + Utd[C, Z] t Xt_dZt + Utd{Ct + [C, Z]t}. If X is the solution of (**), then equating the above with (**) yields =

=

=

=

=

= =

or

dHt Utd{Ct + [C, Z] t}. Since U is an exponential it is never zero and 1/U is locally bounded. Therefore 1-dH dCt + d[C, Z] . * * *) ( t t Ut Calculating the quadratic covariation of each side with Z and noting that [ C, Z], Z] 0, we conclude [ U1 · H, Z] [C, Z]. Therefore equation ( * * *) becomes 1 1 + dH U dC U d[H, Z], and Ct f� U8- 1 d(Hs - [H, Z]s) · Recall that Ut £(Z)t and Xt CtUt, and 0 the theorem is proved. =

=

=

=

=

=

=

=

9

General Stochastic Exponentials and Linear Equations

323

in the formula for £H (Z), it is worthwhile £(Z)"t 1 Z= 1ja continuous £(Z)t appears semimartingale) d[Z, ZJ d ( £(Z)1 ) = dZ -£(Z) and also 1 = £( - Z + [Z, Z]). £(Z) A more complicated formula for £H (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 cad lag, adapted (i. e., H E ]]J)), and let Z be a con­ tinuous semimartingale with Zo = 0. Let Xt = £H (Z)t be the solution of t Xt = Ht + 1 Xs _dZs. Then Xt = £H (Z) t is given by Since to note that (for

= Xt - Ht . Then Y satisfies . t t yt = lo H8_dZ8 + lo Ys _dZs t = Kt + 1 Ys_dZs, where K is the semimartingale H_ Z. By Theorem 52, t 1 yt = £(Z)t{Ko + 1 £(Z) -; d(Ks - [K, Z]s)} and since K0 = 0 and [K, Z]t = J� H8_d[Z, Z]s, Proof

Let yt

·

O+

from which the result follows.

D

Theorem 54 uses the formula of Theorem 52 to give a pretty result on the comparison of solutions of stochastic differential equations.

V Stochastic Differential Equations

324

F

Xt(w)

Lemma. Suppose that is functional Lipschitz such that if = 0, then Let be a continuous increasing > 0 for continuous processes process and let be the solution of

F(X)t - (w) X

X. C

0. Then Pf3 t > 0 Xt < 0} = 0. Proof. Let T = inf { t > 0 Xt 0}. Since X > 0 and X is continuous, Xs > 0 for all s < T on { T < oo } . The hypotheses then imply that F(X) T_ > 0 on 0 {T oo }, which is a contradiction. with x 0 >

:

=

:

0

<

Comment. In the previous lemma if one allows x0 = 0, then it is necessary to add the hypothesis that be strictly increasing at 0. One then obtains the same conclusion. be continuous Theorem 54 (Comparison Theorem) . Let

C

a h<-a-<m (Z semimartingales with Zf) 0, and let Fa be process Lipschitz. Let A be a continuous, adapted process with increasing paths, strictly increasing at t = 0. 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 t t r r Xt = Xo + lo+ G(X) s _dAs + lo F(X)s_dZs, t t = Yo + 1 H(Y) s _dA s + 1 F(Y) s _dZs where > Yo and F and Z are written in vector notation. Then Pf3 t > 0 : Xt < } 0. =

yt

0+

0

xo

Yt

Proof.

=

Let yt ,

Ut = XttNt 1 {F(X)s - - F(Y)s-HXs - Ys) -1 1 {x.;"Ys }dZs, and t r Ct = Xo - Yo + lo+ {G(X)s - - H(Y)s - }dAs. Then Ut Ct + f� Us _dN8, and by Theorem 52 =

=

Next set

9

General Stochastic Exponentials and Linear Equations

325

1

vt = £(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 Wt 0 then G(W£(N) + Therefore K has the property that Wt (w) 0 implies that Y)K(W)t · t- G(Y)t > 0. Note further that K(V) G(U + Y) - H(Y) G(X) - H(Y). Next observe that =

=

=

=

=

=

1

X

where H = (Hi ), 1 < i < n is vector of n semimartingales, takes values is an n x n matrix of adapted, cadlag processes. The processes in JR.n , and 1 < j < m, are given, continuous semimartingales which are zero at zero. Define the operators Fj on l!J)n by

AJ

ZJ,

a

F(X)t A1 Xt =

AJ

where is the n x n matrix specified above. The operators Fj are essentially which process Lipschitz . (The Lipschitz processes can be taken to be

I A11 1

326

V

Stochastic Differential Equations

F(X) t- in the

is cadla,g, not caglad, but this is unimportant since one takes equation.) Before examining equation (*4), consider the simpler system (1

n),

where

8�

=

{0,1,

< i, k <

i = k, i � k.

Letting I denote the n x n identity matrix and writing the preceding in matrix notation yields

t r Ut I + L= Jo0 A� _ Us_dzt, j1 m

=

U takes its values in the space of n n matrices of adapted processes Theorem 55. Let Ai, 1 < j < m, be n 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 n matrix for each ( t, w) and x E JRn , so that is in JRn . If Xf Utx, then since the coefficients are process Lipschitz we Utx 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 J� "L'; 1 A� _ dZt , and therefore the invertibility also follows from Theorem 48. Thus we need to show only that X{ Utx. Since Utx solves ( *4) with Ht x, we have Utx X[ a.s.JRnfor each x. Note that a.s.n the function � U(w)x is continuous from into the subspace of v consisting of xcontinuous functions; in particular ( t, x) � Ut (w )x is continuous. Also as shown in the proof of Theorem 46, (x, t) � X[ is continuous in x and right continuous in t. Since Utx X[ a.s. for each fixed x and t, the continuity permits the removal of the dependence of the exceptional set on x and t. Let u- 1 denote the n n matrix-valued process with continuous trajec­ tories a.s. defined by (U - 1 )t(w) (Ut (w)) -1 . Recall equation ( *4 ) where in ]]J).

x

x

=

=

x

=

Vt

=

=

=

=

=

=

D

X

=

9 General Stochastic Exponentials and Linear Equations

H

and zg

327

[ H , ZJ] [Hi , ZJ].

where is a column vector of n semimartingales = 0. Let denote the column vector of n components, the ith one of which is

Let H be a column vector of n semimartingales, zj (1 < j < ) be continuous semimartingales with ZJ = 0, and let Ai , 1 < j < be n x n matrices of processes in ][)) . Let U be the solution of equation (*5). Then the solution X H of (*4) is given by Theorem 56.

m

m

t j ]s) Xfl Ut Ho + Ut l{o+ U8- 1 (dHs - I:A� d[ H , Z _ j= l Proof Write X H as the matrix product UY. Recall that u -1 exists by Theo­ rem 48, hence Y u - 1 X H is a semimartingale, that we need to find explicitly. m

=

=

Using matrix notation throughout , we have

d(UY) dH + 2:::=l Al_ X_dZj . j Integration by parts yields ( recall that U is continuous) (dU)y_ + U(dY) + d[U, Y] dH + 2:::=l AJ U_ Y_dZJ, j by replacing X with UY on the right side above. However U satisfies ( *5) and therefore i U_Y_dZJ , (dU)Y_ I:A j= l and combining this with the preceding gives U(dY) + d[U, Y] dH, or equivalently dY u-1 dH - u-1 d[U, Y]. Taking the quadratic covariation of the preceding equation with Z, we have d[Y, ZJ ] u- 1 d[H, ZJ], However since U satisfies (*5), since [U -1 d[U, Y], ZJ] 0, 1 < j < m

=

m

=

m

=

=

=

=

=

m.

m

m

j=l

j=l 2::: Al_d[H, ZJJ, j=l m

=

V Stochastic Differential Equations since U equals U Substitute the above expression for d[U, Y] 328

_

we obtain

.

m dY = u -1 (dH - L A-!_ d[H, zi ]), =l

into

( *6) and

j

and since XH = UY, the theorem is proved.

0

1 0 Flows as Diffeomorphisms: The General Case

In this section we study the same equations as in Sect. 8, namely

(Z"h< a <m Z = 0

are no longer assumed to be con­ except that the semimartingales 0. In the general case it is tinuous. For simplicity we still assume that not always true that the flows of solutions are diffeomorphisms of !Rn , as the following example shows. Example. Consider the exponential equation in IR,

Z

Z

Z0 =

Let be a semimartingale, 0, such that has a jump of size 1 at a stopping t ime T, T > 0 a.s. Then all trajectories, starting at any initial value become zero at T and stay there after T, as is trivially seen by the closed form of the solution with initial condition

x,

-

x:

Therefore, injectivity of the flow fails, and the flow cannot be a diffeomorphism of R We examine both the injectivity of the flow and when it is a diffeomorphism of !Rn . Recall the hypotheses of Sect. 7, to which we add one, denoted (H3). Hypothesis (Hl) . are given semimartingales with 0, 1 < a < m . : !Rn IR are given functions, 1 < i < n, 1 < a < m, Hypothesis (H2).

and f(x)

Z" f� denotes the n

Z[j' =

--+

x m

The system of equations

matrix (f�(x)).

m t �(X8_)dZ�, x; = xi + L Jo f a=l 0

1
10

Flows as Diffeomorphisms: The General Case

329

may also be written

!Rn , f(Xs- ) is an n

Xt x

where and are column vectors in is a column vector of m semimartingales.

xm

matrix, and Z

Hypothesis (H3). f is c= and has bounded derivatives of all orders. Note that by Theorem 40 of Sect. 7, under (H3) the flow is c=. 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 we can find stopping times 0 = tending a.s. < < < to oo such that (Z"'fi- - Z fi have an norm less than E (cf. , Theorem 5). Note that by Theorem 1, are smaller than E. < E as well, hence the jumps of each occur only at the times (Ti ) , j > 1. Therefore all of the "large" jumps of Let xt denote the solution of *) driven by the semimartingales Outside of the interval the solution is

T0 T1 T2 z a ,J = ( "' 1 ,j a z za ,j ( (Tj - l, T1 )

(Z"'h
·

·

·

-

H, 00 [za,j , za j ]�2 (X)

za ,j .

if t < Ti - l , if t > Ti .

Next define the linkage operators

using vector and matrix notation. We have the following obvious result. Theorem 57. The solution of (*) is equal to, for
Tn

X

where

nl

,

xx + = xXf (x), H1 (x 1 -) 1x -- = Xf,1 __ (x i+), xx 1 ++ == H2(x 2 -) 2 2 2 Xn- = XTn _ (X( n- 1) + ), Xn+ = Hn (Xn - ) · Theorem 58. The flow Xt (x, w) of the solution X of (*) zs a x diffeomorphism if the collections of functions x Xf (x, w) o

=

cp

f---*

are diffeomorphisms.

:

--.

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 0 morphisms is a diffeomorphism, the theorem is proved. We begin by studying the functions X __.. x? (x, w) and X __.. x� + 1 (x, w). Note that by our construction and choice of the times T1 , we need only to consider the case where Z = zi has a norm in H00 smaller than E . The following classical result, due to Hadamard, underlies our analysis. Theorem 59 (Hadamard's Theorem) . Let g : !Rn -.. !Rn be coo . Suppose J

(i) lim ll x ll-+oo ll g(x) ll = oo, and (ii) the Jacobian matrix g'(x) is an isomorphism of !Rn for all x. Then g is a diffeomorphism of !R n . Proof By the Inverse Function Theorem the function g is a local diffeomor­ phism, and hence it suffices to show it is a bijection of !Rn . To show that g is onto (i.e. , a surjection), first note that g(!Rn) is open and non-empty. It thus suffices to show that g(!Rn) is a closed subset of !Rn , since !Rn itself is the only nonempty subset of !Rn that is open and closed. Let ( xi )i> 1 be a sequence of points in !R n such that lim i-+oo g(xi ) = y E !Rn . We will show that y E g(!Rn ). Let Xi = tivi , where t i > 0 and I vi I = 1. By choosing a subsequence if necessary we may assume that Vi converges to v E sn , the unit sphere, as i tends to oo. Next observe that the sequence (t;)i> l must be bounded by condition (i) in the theorem: for if not, then t i = I xi I tends to oo

along a subsequence and then llg(xi k ) ll tends to oo by (i), which contradicts that limi-+oo g(xi ) = y. Since (ti)i> l is bounded we may assume limi-+oo ti = t o E !Rn again by taking a subsequence if necessary. Then limi-+oo xi = tov, and by the continuity of g we have y = limi-+oo g(xi ) = g(tov). To show g is injective (i.e., one-to-one) , we first note that g is a local homeomorphism, and moreover g is finite-to-one. Indeed, if there exists an infinite sequence (xn ) n > l 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 Xn tends to x (the cluster point), where g(xn) = yo all n. By the continuity of g we have g(x ) = y0 as well. This then violates the condition that g is a local homeomorphism, and we conclude that g is finite-to-one. Since g is a finite-to-one surjective homeomorphism, it is a covering map. 16 However since !Rn is simply connected the only covering space of !Rn is !Rn (the fundamental group of !Rn is trivial). Therefore the fibers g-1 (x) for x E !Rn 0 each consist of one point, and g is injective. ,

The next step is to show that the functions x �---' X� (x, w) and x �---' x� + 1 ( x, w) of Theorem 58 satisfy the two conditions of Theorem 59 and are J

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 injectivity which is closely related to Theorem 41.

Let za be semimartingales, 1 < a < m with Z0 = 0, and let F be an n x m matrix of process Lipschitz operators with non-random Lipschitz constant K. Let Hi E IDl, 1 < i < n (cadlag, adapted). If'f:.r;: 1 I Za l -w"' < for > 0 sufficiently small, then the flow of the solution of Theorem 60.

c: ,

c;

is weakly injective. 1 7 Proof Let x, y E !Rn , and let xx , XY denote the solutions of the above equa­ tion with initial conditions x, y, respectively. Let u = x - y, U = x x - XY, and

V = F(X x) _ - F(XY)_ . Then V E L and \ V I < K I U- 1. Also,

!::.. Us 'f:.a V8a t::..zr; li t::.. Us I < l �a l l t::..zr; I < CI I Us- l c < 21 1 \ Us - 1 if c is small enough. Consequently 1 \ Us l > � I \ Us - 1 · Define T = inf{t > 0 : 0}. Then Ut - =/= 0 on [0, T) and the above implies Ut =/= 0 on [0, T) as Utwell.- =Using Ito's formula for f(x) = log l x l , in the proof of Theorem 41 we have

= and moreover (using the Einstein convention Therefore to leave the summations implicit)

as

For

s fixed, let

1 7 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 Js . By Taylor's Theorem

(

l.::o <s
f(x)

= log where Us - · Since

1 / x l , and Is denotes the segment with extremities Us and 2f 8 sup I :.::. :.::. . (x)l < sup ( Z,J uX,uX J X 11 2 ) ' I Els x Els 1

x and since II �Us I < � IIUs - 11, we deduce .

which in turn implies

Therefore

[0, T). Returning to ( ** ) , as t increases to T, the left side tends to -oo on { T < oo} and the right side remains finite. Therefore P(T < oo) = 0, and U and U never vanish, which proves the theorem. 0 Theorem 61. Let (Z"'h< a < m be semimartingales, Z(f = 0, F an n x m matrix of process Lipschitz operators with a non-random Lipschitz constant, and Hi E ][)) , 1 < i < n. If 2:;:' I I Z"' I < E for E > 0 sufficiently small, then for r E IR there exist uniformly locally bounded predictable processes H"' (x,y ) and K"'·f3 (x ,y), which depend on r, such that II X x - XY II'" = l x - YI I '"E (A,. (x, y)) where x x is the solution of on

_

1

The semimartingale A,. is given by

H'"'

10 Flows as Diffeomorphisms: The General Case

333

Lo <s<- t As , and where As is an adapted process such that l As I < Cr ( � Z�)2. Proof Fix x, E !Rn , x =/= and let U = xx - XY , V = F(X x )_ - F(XY)_ . By Theorem 60 U is never zero. As in the proof of Theorem 43, by Ito's where Jt

=

y

y,

formula,

t I Uy l r = l x - Yl l r + lo r i J Us - l r- 2 u; _du; t + � lo r{ (r - 2) I Us - l r- 4u;_ u;_ + 8j I Us - l r- 2 }d[ Ui ' Uj ] � + <sL
Is is the segment with boundary Us and Us - · However,

in the proof of Theorem 60, we However 2• C x l l r r which implies ( for all positive or negative) that 1 , Us � Us I I l xl l r < Cr i Us - l r for all x between Us - and Us. Hence (the constant C lchanges) I Lsl < C J � Usii 2 I Us -l l r-2 < C J Us - l n J �Zsll 2 . Therefore, let As = I Us - 1 - r L s, and we have which is less than < � have

as

r

an absolutely convergent series with a bound independent of plete the proof it suffices to take

and

(x, y). To com­

334

V Stochastic Differential Equations

K�'f1(x, y) = � r{(r - 2)\ \ Us-I \ 4 ( Us _, vsa )(Us - , Vf ) + I \ Us - I - 2(Vsa , vf )}1 {Us-#O} , as in the proof of Theorem 43. Note that 1{u, _ ,.oo} is indistinguishable from the zero process by weak injectivity (Theorem 60) . These choices for H a and are easily seen to work by observing that Uf = 2::;; 1 J; �i ,a dz<; , and Kthea ,/1preceding allows us to conclude that t \ Ut\ 1 " = 1 \ x - Yl l " + lo \ Us - \ " dA,. (x, Y)s, 0

and the result follows.

Let (Za h< a < m be semimartingales, Zff = 0, F an n x m matrix of process Lipschitz operators with a non-random Lipschitz constant, and H i E ][)) , 1 < i < n. Let X = X(t,w , x) be the solution of

Theorem 62.

t

Xt = X + lo F(X)8_dZ8 • If 2:;: 1 1 \ za \1 almost all w

H'"' -

< E

for E > 0 sufficiently small, then for each N < oo and

( s , x) \ 1 = 1 \ X < s llxll-+oo N Proof. The proof is essentially the same that of Theorem 45, so we only which is well-defined by weak sketch it . For x =/= 0 let yx = I xx X0 I w,

lim inf

oo.

as

-

injectivity (Theorem 60) . Then

-l,

\ Yx YY j 1 \ Xx XY \ 1 1 \ X x XO \ I -1 1 XY X0 \1 - 1 = 1 \ x - yl \ l x l \ - 1 1 \ y i -1 E(A 1 (x, y ))E( A -1 (x,O))E( A-1 ( y,O)) by Theorem 61, where A,. (x, y) is as defined in Theorem 61. Set yoo = 0. Since 1 \ Z I H= each za has jumps bounded by and the process lt defined in Theorem 61 also has jumps bounded by C,.c2 . Therefore we can stop the processes A,. (x, y) at an appropriately chosen sequence of stopping times ( Te)e> 1 increasing to a.s. such that each A ,. (x, y) E S(c) for a given and for each £, uniformly in (x, y ). However if Z is a semimartingale in S(c) , then since E(A,. (x, y)) satisfies the equation Ut = 1 + J;+ U8 _dA,. ( x, y)8, by Lemma 2 of Sect. 3 of this chapter we have I E (A,. (x, y)) l \ sv < C(p, z) _

<

_

_

_

< E

E,

E,

oo

< oo,

10

Flows

as

Diffeomorphisms: The General Case

(,

335

= I A,. (x, y)l l w"'

< ke, the where C p z) is a constant depending on p and bound for £, provided of course that c is sufficiently small. We conclude that for these Te there exist constants Ce such that z

where p > n, and where d is the distance on !Rn \ { 0} given by d( Set 0< < oo,

,yx = { yyooxllxl -2 '

( xC x (Yx ) -1 = I Xxl l xl - 2 - X0 J I ,

=0

'

lxl l x l = 0.

x, y) = i11;1�1�111 .

Then II Y YIIP on !Rn , and by Kolmogorov' s Lemma Y Y fe ll�v < fll (Theorem 72 of Chap. IV) we conclude that lim l i i O y x exists and it is zero. the result follows. 0 Since -

l x -+

If cp is the flow of the solution of ( *), Theorem

62 shows that

= +oo, cp(x) l l l l xl -+oo lim

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 let be an n x n matrix of given semimartingales. Recall that denotes the (matrix-valued) exponential of and that de­ notes the (matrix-valued) right stochastic exponential of which was defined in Sect. 8, following Theorem 47. Recall that in Theorem 48 we showed that if is an n x n matrix of continuous semimartingales with + or 0, then The general case is more delicate. equivalently +

n, Z

[R (z ) Z, Z, Z R = Z [ Z , Z]) = I, ( E(Z)E Z0 E( -Z [Z, Z])E R (z) = I. Theorem 63. Let Z be an n n matrix of semimartingales with Z0 = 0. Suppose that Wt = Zt + [Z, Z] � + Lo<s<- t (I + � Zs) -1 (� Z8)2 is a welldefined semimartingale. Then X = E (Z)

x

-

for all t > 0. Proof Let U Then,

and therefore

= E(W), V = [ R (z), and lt = Lo<s
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, Z]V_ = -U_d[Z, Z]V_ + U_d[J, Z]V_, since d[Z, [Z, z]c] = 0. By Theorem 47 d(UV) U_dV + (dU)V_ + d[U, V] ;

=

using the above calculations several terms cancel, yielding

d(UV) = U_d[Z, Z] cv_ + U_(dJ)V_ - U_d[Z, Z]V_ + U_ d[J, Z]V_.

the preceding becomes

d(UV) = -U_d(I:: (�Z8)2)V_ + U_d( I:: (� Z8)2(I + �Zs) 1 )V_ + U_d( I:: ( flZ8 )3 (I + �Zs) -1 )V_. Since ( �Zs) 2 ((�Z8)2 + (�Z8) 3 ) (J + flZ8 ) - l , the above equation implies d(UV)t = 0, all t > 0. However U0 Vo I, and therefore Ut = I, all t > 0. Corollary. Let Z be a square matrix of semimartingales. If I Z I w"' < E for E > 0 sufficiently small, then [R (Z)t is invertible for all t > 0. Proof If I Z I H = < c then the jumps of Z are bounded by E > 0 and therefore the process of Theorem 63 is a well-defined semimartingale. Theorem 64. Let (Z"'h< a < m be semimartingales with Z0 = 0, and let f be a matrix of coefficients satisfying Hypotheses (H2) and (H3). Let X be the -

·

=

vt

W

0

0

unique solution of

The Jacobian matrix Dic(t, x ) = 8�k Xi (t, x ) is invertible for each t > 0 provided I Z I H= < E for sufficiently small E > 0. w,

w,

10 Flows as Diffeomorphisms: The General Case Proof By Theorem 39 (in Sect. 7) the Jacobian matrix

stochastic exponential equation

337

D satisfies the right

t {}fi 1 Dt = I + ( , (Xs- )dZ�)Ds- , 0

{)

Xk

and the matrix semimartingale differential ��: (X8_ )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 !Rm ; recall that under Hypotheses (Hl ), (H2), and (H3), that Z ( Z, h < n < 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 !Rm H(x) = X + f(x)z is a diffeomorphism of !Rn } I = {z E !Rm H(x) = X + f(x)z is injective in !Rn }. :

:

Clearly V c I.

Let Z and f be as given in Hypotheses {Hl), (H2), and (H3), and let X be the solution of t Xt = X + f(Xs - )dZ8•

Theorem 65.

1

The flow of X is a.s. a diffeomorphism of !Rn (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;' +1 (x, w) defined in Theorem 57, and the linkage operator Hi (x) = x + f(x)Ll Zr; , define� immediately preced­ J

ing Theorem 57. By hypothesis the linkage operators flJ (x) are clearly diffeo­ morphisms of !Rn (resp. injective), and by Theorems 62 and 64, Hadamard's conditions are satisfied (Theorem 59) , and therefore the functions x X? (x, w) and x X;' + l (x, w) are diffeomorphisms of !Rn if c > 0 is taken small enough in the definition of the stopping times (TJ )j > 1 , which it is always possible to do. Therefore by Theorem 58 the flow c.p x Xt(x, w) is a.s. a diffeomorphism of !Rn for each t > 0. 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 llf(x)zll < llxll /2 since f is Lipschitz, which implies that II H(x)ll > llxll/2 and thus condition (i) of Hadamard's Theorem is satisfied. On the other hand H'(x) = I + f'(x)Z is invertible for all x for llzll 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. r-t

J

r-t

:

-+

V Stochastic Differential Equations

338

To prove necessity, set

F1 = {w : 3 > 0 with �Z8(w) E vc} , F2 = {w : 3 > 0 with �Z8(w) E Ic} . Suppose P(F1 ) > 0. Since V contains a neighborhood of the origin, there exists an c > 0 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---> Xj.i ( x) are diffeomorphisms as soon as the linkage operators Hk 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 Ai = {w : �Zri E V, all i < j, and �Zri E vc} . Since P(F1 ) > 0, there must exist a j such that P ( Ai ) > 0. Then for w E Ai , w) is a diffeomorphism, but Hi (x, w) is not a diffeomorphism. xLetf--->woX E _A1(x,and to T1 (wo). Then x f---> Xt0 (x , wo) is not a diffeomorphism, s s

Ti

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 D have injectivity is analogous. Corollary. Let Z and f be as given in Hypotheses (Hl), (H2), and (H3), and let X be the solution of

Xt = X +

t

1 f(Xs - )dZ8•

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 T1 that slice the semimartingales za into pieces of H= norm less than c. If the za do not jump at Tio for some ]o, however, and paths of X intersect there, one need only slightly alter the construction of Tj0 (cf. , the proof of Theorem 5, where the times T1 were constructed) , so that Tjo is not included in another sequence that c-slices (Zn h< a < m , to achieve a contradiction. (Note that if, however, (Zn h< a < m has a large jump at Tj0 , D then it cannot be altered.) 1 1 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

Lemma. For any predictable (matrix-valued) process and for 0 < t < 1 , we have

339

H and for any p > 1 ,

Since 0 < t < 1 , we have that ds on [0, 1] is a probability measure, and since f(x) = xP is convex for p > 1 , we can use Jensen's inequality. The result then follows from Fubini's Theorem. 0

Proof

Lemma. For any predictable (matrix-valued) process sional Brownian motion B, and for any p > 2, and for 0 a constant Cp depending only on p such that

H and multidimen­ < t < 1 , there exists

The proof for the case p = 2 is simply Doob's inequality; we give it for the one dimensional case.

Proof

where the last equality is by Fubini's Theorem. For the case Burkholder's inequality (see Theorem 48 of Chap. IV,

p>

2, we use

where the last inequality follows from Jensen's inequality (recall that p > 2 so that f(x) = xP/ 2 is a convex function, and since 0 < t < 1 we have used that ds on [0, 1 J is a probability measure) and Fubini's Theorem. 0 Theorem 66. Let Z be a d-dimensional Levy process with E{ll Zt I I LP } for 0 < t < 1 , H a predictable (matrix-valued) process and p > 2. Then exists a finite constant Kp such that for 0 < t < 1 we have

< oo there

Proof We give the proof for the one dimensional case. 1 8 Since the Levy pro­ cess Z can be decomposed Zt = bt + aBt + 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 = aBt > and Zt = Mt . 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] .

V Stochastic Differential Equations

340

Cp

vary Kp(·) < < 2k + 1 .

In the computations below, the constants and the functions from line to line. Choose the rational number k such that 2 k p Applying the Burkholder-Gundy inequalities 1 9 for p > 2 we have

Set

= oo. 2 } vM(dx) Msl aM := E{[M, M] I} = E{L J i x l 2 I � < s< l Since [M, M] is also a Levy process, we have that [M, M] t - a M t is also a martingale. Therefore the inequality above becomes p p t tr /2 r E{ Jo HsdMs }
We apply a Burkholder-Gundy inequality again to the first term on the right side above to obtain

p t / M H s 4} E{ lo HsdMs } < CpE{(L si 4 )P � I s< t t + Kp( t )[J l x l 2 vM (dx)] PI2 E{ 1 I Hs i P ds}.

We continue recursively to get

t P k k + l l} / 2 + P 2 E{ lo HsdMs } < CpE {( L s�Ms ) H I l s
Next we use the fact that, for any sequence a such that llal lzq is finite, ll a llz2 l l a iiz q for 1 q 2. As 1 p2- k 2 we get

<

whence 19

< < < < k+ l JPI2k+ l = [L U Hs�Msl 2k )2J h1< 2 [L s�Msl H I s
The Davis inequality is for the (important) case p = 1 only.

1 1 Eclectic Useful Results on Stochastic Differential Equations

341

s
t

; liM (dx) )PT ; ] 2 x E{ l 1 HsdMs iP } <[Cp J l x iP vM (dx) + Kp(t) I) l / l •=1 t E{ fo I HsiPds} It remains to show that, for any 1 < i < k, (j l x l 2; VM (dx))PT ; < (j lx l 2vM (dx))PI2 + J lx iPvM (dx). Let AM := J l x l 2 vM(dx), so that 0

·

is a probability measure. Denote

If

2i by q. One has to show

p fq q fq 2 2 ) x x Z < (d )P A t (j l l �- M

the preceding inequality is obvious. On the other hand, if

then it is sufficient to prove that

But the bound on

AM and Jensen's inequality give the result.

D

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 cr be continuously differentiable with a bounded derivative,

and let Z be a Levy process with Levy measure v such that � x i>I lx i Pv(dx) < which is equivalent to E{ IZtiP} < for p > 2 and for all t, 0 < t < 1. Let xx denote the unique solution of the stochastic differential equation

oo ,

oo

dXt = cr(Xt - )dZt , Xo = x. Then we have

IX_;'I P} < K(l + lxi P) E{sup s 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) .

itself, and suppose

a

Let a be a function from IR + to

s (s) < c + k 1 a(r)dr <

oo

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.

inequality.

This is a consequence of Theorem 66 and Gronwall's 0

The next corollary follows easily from Theorem 67. Theorem 69. q > 0 define

Let cr and Z be as in Theorem 67. Let g be continuous and for

II g ll q = inf{a > 0 : lg(x) l < a( l + lxl q )}. For q = 0 we let I I g ll o be the L00 norm. If Pt denotes the transition semigroup of the Markov process x x 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 v and with finite p-th moment for p > 2, we decompose it as Zt = bt + cBt + Mt where B is a standard Brow­ nian motion and M is a martingale independent from B and L2 orthogonal to all continuous martingales. We then define

which is finite since by assumption E{IZtiP} < oo and p > 2. Here we consider the multidimensional case: both Z and the solution x x of equation ( * ) are assumed to be d-dimensional, where d > 1 . 2 1 We let x: · ( k ) represent a k-th derivative of the flow of xx . Theorem 70. For k > 1 assume that cr is differentiable k times and that

all partial derivatives of cr of order greater than or equal to one are bounded. Assume also that Z has moments of order kp with p > 2. With the nota­ tion defined right before this theorem, we have that there exists a constant K(k,p, cr, 'T/p) such that for 0 < s < 1, E { sup 1 x; , ( k) I P} < K. 8

We recall from Theorem 39 that x x together with the unique solution of the system of equations

Proof

xx, ( k)

constitute

(D) as long as k > 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

Xtx ,(k)

1 'Vcr(Xx )Xx,(k) dZ t

=

0

8-

s-

8

if k ;:::: 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

t

xx ,( l )

=

I +

d

10t 'Vcr(Xx )Xsx-,( l) dZs . s-

We next observe that the components of Fk , i (x< 1 l, . . . , x< k -i + l ) ) are sums of terms of the form

k -i+ l II j =l

C>j II r=l

where

L jai J

=

( ** )

k,

and where xU ) , l is the l-th component of xU) E 1Rd1 , and an "empty" product equals 1. We thus want now to prove that

t E{sup 1 'Vi cr(Xs'x- )F t

p

· (Xsx-,( l) ' . . . ' Xsx-,( k -i+ l ) )dZs } <- K

N,z

0

K(k,p, cr, 'T/kp) (in the remain­ for all i 2, . . . , N and for some constant der of the proof K K(k, p, cr, 'T/kp) varies from line to line) . And of course, it is enough to prove that if G is any monomial as in ( ) then

K

=

=

=

**

We will use Theorem 66 and the fact that rem 66 implies that

,

'V i er is bounded. Note that Theo­

for 2 < p' < p, where the constant K(p, 'T/p ) depends only on p, 'T/p , and the dimensions of H and Z. For this we see that the left side of the previous equation is smaller than

k +l -i k+ l- i K E{ IT sup IX: , (j ) l paj } < K II (E{sup IX�x , (j ) l kpfj } )] nj /k s j=l J. = 1 s by Holder's inequality, since L:i ja1 k. The recurrence assumption yields that each expectation above is smaller than some constant K(p, k, cr, 'T/kp), 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.

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 t 1 1 Eclectic Useful Results on Stochastic Differential Equations

Xt = Xo + 1 cr(Xs )XsdZs with X0 > 0 almost surely. Let T = inf {t > 0 : Xt = 0}. Then P(T < ) = 0. In words, if the coefficient of the equation is of the form xcr (x) with cr ( x) continuous on [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 oo

oo ,

the applied literature the equation is often written in the form

= x �x) , but Note that if cr(x) is any function, one can then write 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 Since being Lipschitz continuous is a sufficient condition to have a unique and non-exploding solution, we can require in turn for to be both bounded and Lipschitz, which of course implies that xcr(x) is itself Lipschitz.

cr(x) ( ) ""

xcr(x)

cr

cr.

Proof of Theorem 71. Define Tn = inf{t > 0 : Xt = 1/n or Xt = Xo 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

cr

and since is assumed continuous, it is bounded on the compact set [1/n, n] , say by a constant c. Since the stopping times Tn j T we see that on the event {T < oo} the left side of the above equation tends to oo while the right side 0 remains finite, a contradiction. Therefore P (T < oo ) = 0.

Z

Theorem 71 has an analogue when is no longer assumed to be continuous. Theorem 72. Let be a semimartingale and let be a bounded, continuous

cr Z function such that there exists a unique non-exploding solution of t Xt = Xo + 1 cr(Xs_)X8_dZ8 with Xo > 0 almost surely. If in addition we have I� Zs I < l �li: almost surely for all s > 0 for some c with 0 c 1, then the solution X stays positive a.s. for all t > 0.

<

<

346

V Stochastic Differential Equations

Proof II a lloo refers to the L00 norm of cr. of Theorem 71. Using Ito's formula gives

The proof is similar to the proof

t{

I ti ) = ln(Xo) + lot cr(Xs-)dZs - �2 Jo cr(X8_) 2d[Z, Z]� + L { In IXsl - ln I Xs - 1 - cr(Xs - )�Z8}, s
ln ( X

and let us consider the last term, the summation. We have

L { In I Xsl - ln IXs - 1 - cr(Xs - ) �Zs}

s
where IHsl < the above is

1

by the Mean Value Theorem, applied w-by-w , which implies

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 ( - oo , OJ at time s-. Then we have I�Xsl < Xs- l cr(Xs- )�Zsl < Xs - ( 1 - c) < Xs_ , which implies that X cannot cross into ( - oo , OJ by jumping. Hence we have that X must be positive on [0, oo ) . 0 We now turn to something completely different. In Sect. 6, Theorem 32, we showed that an equation of the form

t Xf = Xo + L 1 f} (Xs - )dZi 0 d

j=l

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

Let (D, F, lF, P) be a filtered probability space with Z a cadlag semimartingale defined on the space. Let f be a Borel function which is never 0 and is such that for every x E lR the equation Theorem 73.

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

D' be the function space of dtdlag functions defined on JR+ with Xt(w) = w (t) , the projection operator. Let JF' be the canonical filtration and let (en be the shift operators. If p� denotes the law of xx' then we have that ( D' ' JF' ' ei, X I' PD is a Markov process in the Dynkin (or Blumenthal-Getoor) sense. Since f is never zero, we can write Let

We can therefore define on D' , relative to each law

P� , the stochastic integral

Z' = 1t f(Xs' - ) 1 dX' t 0 and we have that the law of Z' under P� is the law of the process Z - Z0 . However Z' is also an additive functional, and hence the Markov property -

S'

implies

where we have also used that the law of Z' under P� is the law of the process Z - Zo. 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 - Z0 . Using the identity of the laws of Z' under P� with that of the process Z - Z0 once again, the result follows. D Bibliographic Notes

The extension of the HP norm from martingales to semimartingales was im­ plicit in Protter [197] and first formally proposed by Emery [64] . 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 [197]. 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 [81] in this regard. These results were improved and simplified by Doleans-Dade-Meyer [54] and Emery [65]; our approach is inspired by Emery [65] . Metiver-Pellaumail [161] have an alternative approach. See also Metivier [158]. Other treatments can be found in Doleans-Dade [52] and Jacod [103]. The approach of L. Schwartz [214] is nicely presented in [44]. The stability theory is due to Protter [199], Emery [65] , and also to Metivier-Pellaumail [162]. 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 [ 171]. The treatment here of Fisk-Stratonovich differential equations is new. The idea of quadratic variation is due to Wiener [230]. Theorem 18, which is a random Ito's formula, appears in this form for the first time. It has an antecedent in Doss-Lenglart [58], 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 [92] and Sussman [221]). The treatment of approximations of the Fisk-Stratonovich integrals was inspired by Yor [238]. For an interesting ap­ plication see Rootzen [211]. For more examples of solutions of stochastic dif­ ferential equations with formulas, see [128]. The results of Sect. 6 are taken from Protter [196] 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, 210]. Work on flows of stochastic differential equations goes back to 1961 and the work of Blagovescenskii-Freidlin [18] 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 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,

,

L2

Exercises for Chapter V

349

native proof is in Uppman [225] . For the general (right continuous) case, we follow the work of Leandre [141] . A similar result was obtained by Fujiwara­ Kunita [78] . Our presentation on moment estimates follows [205] and [105] , although it dates back to [16] . A perhaps simpler treatment using random measures can be found in [105] . See also [106] . 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 [108] . Exercises for Chapter V

X be a solution of the stochastic differential equation dXt = X'fdBt + X(dt with initial condition X0 f:. 0. Show that X is defined on [0, T) , where T = inf{t > 0 Bt = 1/X0}. 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 = 0. Let X be a solution of the equation Exercise 1 . Let

:

with initial condition X0. 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 X0 given, and a and b both Lipschitz continuous. Let the generator of the (time homogeneous) Markov process X be given by

Ptf(x) - f(x) Af(x) = t -+lim O , t>O t where Ptf(x) equals the transition function2 2 P8 (X0 , f) when X0 = 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 f 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 is a linear partial differential operator for functions E C� given by

X

f

Af(x) =

fP 2 a (x ) &x2 2

f + b(x) &af .

1

x

Exercise 5. Let E Ct, let a and b be Lipschitz continuous, let unique solution of the stochastic integral equation

f

X be the

and let denote its infinitesimal generator. If = x , where x E JR, prove Dynkin's expectation formula for Markov processes in this case:

A

Xo

t

E{f(Xt) - f(Xo ) - lo Af(Xs)ds} = 0. Exercise 6. As a continuation of Exercise 5, show that

t (Pd)(y ) - f(y) - lo (PsAJ)(y)ds = 0 for y E JR, and that lims! O P8Af = Af for (for example) all f in compact support.

c=

with

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 E CC and a and b are both Lipschitz, and is a solution of equation * ) of Exercise 5, then

(

f

X

( T f(Xr) - f(Xo) - 1 AJ(Xs)ds is a local martingale. Exercise 8. Let Z be a Levy process with Levy measure and let X satisfy the equation t Xt = X + lo f( Xs_)dZs where f is Lipschitz continuous. The process X is taken to be d-dimensional, and Z is n-dimensional, so f takes its values in JRd x JRn. We use (column) vector and matrix notation. The initial value is some given E JRd . Let p x denote the corresponding law of X starting at Show that if g E with d compact v

support on JR , then

x.

x

c=

Exercises for Chapter V

351

( ) .. g Ag(x) = 'Vg(x)f(x)b + 2::: &xi &xi (x) (f(x)cf(x)* )'1 + j v(dy) (g(x + f(x)y) - g(x) - 'Vg(x)f(x)) , where 'Vg is a row vector and b is the drift coefficient of Z, denotes transpose, and A is of course the infinitesimal generator of X. 2

d

1

"'2

u

.

< ,J=l

*

Exercise 9. In the setting of Exercise 8 show that

g( Xt) - g(X0) -

t

lo Ag(X.)ds is a local martingale

for each law px . Show further that it is a local martingale for initial laws of the form PJ.J., where PJ.J. is given by: for A C D measurable, PJ.J.(A) = JIRd px (A)J..L (dx) for any probability law J..L on JR d . Exercise 10. Let be an arbitrary semimartingale, and let T = 0 and denote the increasing sequence of stopping times of the jump times of when the corresponding jump size is larger than or equal to 1 in magnitude. Define = sign 6. rJ with = 1 . For a semimartingale y define Y)t to be the solution of the exponential equation

(ZTn )n>

1

En

o

Z

(Z

un (

Eo

t { Vt = 1 +

n t

v;,_d rJ n Ys

for T < < Tn + I · Show that the (unique) solution of the equation

t

1 IXs- l dZs L::n ->o Xf1[Tn,Tn+ d(t), where xr Xrn un (En Z)t, for Tn < Xt = 1 +

= is given by Xt = < Tn + l · Exercise 11. Show that if M is an L2 martingale with 6.M8 > -1, and (M, M)00 is bounded, then £(M)00 > 0 a.s. and it is square integrable. (See [14 7].) Exercise 12. Let M be a martingale which is of integrable variation. Suppose 16.Ms I is bounded. Show that also that the compensator of the process the local martingale E(M) is also of integrable variation. Exercise 13. Let be a semimartingale with decomposition = L + V where L is a local martingale with 6.L8 of. - 1 for all finite s , and V has paths of finite variation on compacts. Let

t

L::s
Z

Show that E(Z)

=

E(L)E (U) . (See [157] .)

Z

352

Exercises for Chapter V

*Exercise 14 (Memin's criterion for exponential martingales) . Let M be a local martingale, let It = L s ! } ' and let Lt = Jt - It and Nt = Mt - Lt . (Both L and N are of course also local martingales. ) Let

At = [M, M] c + L i �Ls i + L (�Ns ) 2 O<s
Exercise 15 (Gronwall's inequality) Let Ct be a ca.cllag increasing pro­ cess and suppose 0 < At < a: + J� As - dC8 for t > 0. Show that At < ae 0• , each t > 0. ( Note: The classical Gronwall inequality is usually stated with Ct = t. Also see [85] .) The next five exercises outline the Metivier-Pellaumail method for showing the existence and uniqueness of solutions of stochastic differential equations. •

.

*Exercise 16. Let M be an L 2 martingale. Show that for any predictable stopping time T, E{(E{�Mr iFr - }) 2 } < E{(M, M) r_ }. Show also that for any totally inaccessible stopping time T, we have E{(E{�MriFr - }) 2 } < E{ (M, M) r }. Conclude that for an arbitrary stopping time T we have

E{(E{�Mri Fr- }) 2 } < E{( M, M)r_ }. Exercise 1 7 (Metivier-Pellaumail inequality) . Let M be an L 2 mar­ tingale and let Mt_ = sup s
that for any stopping time T we have

Mo)2} < 4E{( M, M)r_ + [M, M] r - }. s E{ sup(M s< T Exercise 19 (extended Gronwall's inequality) . Let A and C be two increasing processes with A adapted and E{A oo } < oo, and C such that 0 < C00 < k everywhere for some constant k. Suppose that for every stopping

time T and for some positive constant a: we have

Then E{A oo} < ae k . ( Hint: Use Lebesgue's change of time lemma, together with Gronwall's inequality of Exercise 15. Also see [110, page 576] . )

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 2 { t = 3 Xl + 3 Xt Jo Xl ds Jo dB8 , Xo = 0, 1

B is a standard Brownian motion. (See [121, page 293].) *Exercise 22. Let a satisfy a weakening of the Lipschitz condition of the form ia(y) - a(x) i < 11;(ix - yi) for all x, y E lR, with J; ,.. �)2 du = and 11; ( increasing with 11;(0) = 0. (Such a condition is a variant of a standard condition

where

oo

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

B

where is standard Brownian motion. (See [232] . ) Exercise 23. Let and be Lipschitz continuous and let be the unique < c + k lxl 2 , solution of equation (*) of Exercise 5. Show that 2xb(x) + where c and k are positive constants. Use Ito ' s formula on to show the < following + c ekt . (See [135] .) *Exercise 24 {Euler method of approximation). Let and be con­ tinuous and such that a unique path-by-path solution X of equation ( *) of Exercise 5 exists. Let (�k ) k >I be i.i.d. with mean zero and variance r2 . Let Xo be independent of (�k ) k >I and let xk be defined inductively by

a b

moment estimate: E{Xl} (E{X5} t) Xk+ l -_ Xk +

�k +l r.:: + (}' yn

X a(x)2 Xf a b

b(Xk) n

Xt n (t) = X[nt] [where [nt] denotes the integer part of nt. Let Bn (t) = Jn It J1 �k , Vn (t) = ":] , and finally observe that t t Xn (t) = X(O) + 1 a(Xn (s-))dBn (s) + 1 b(Xn (s-))dVn (s). By Donsker's Theorem we know that (Bn , Vn ) converges weakly (in distri­ bution) to ( B , V) , where B is a standard Brownian motion and V(t) = t. Show that Bn is a martingale, and that Xn converges weakly to X. (See [136] and [137] .) Define

r

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 xt 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 g with at most linear growth, set Assume that f is infinitely differentiable, and that g is twice differentiable, bounded and both of its first two derivatives are bounded. Show that the function (t, x) 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 f--4

VI Expansion of Filtrations

1 Introduction

By an expansion of the filtration, we mean that we enlarge the filtration (Ft)t > O to get another filtration (1it)t> O such that the new filtration satisfies the usual hypotheses and Ft C 1it, each t > 0. 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 (Ft)t>o of B by adding the a-algebra generated by the random variable B 1 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 is a positive random variable, and one expands the filtration in a minimal way to make 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.

L

L

356

VI Expansion of Filtrations

2 Initial Expansions

Throughout this section we assume given an underlying filtered probability space (r!, F, (Ft)t>o, P) which satisfies the usual hypotheses. As in previous chapters, for convenience we denote the filtration (Ft)t > o by the symbol lF. The most elementary result on the expansion of filtrations is due to J acod 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 A a , A,a E A then Aa n A,a = ¢, a: of (3. Let 1it be the

filtration generated by Ft and A. Then every ((Ft)t>o, P) semimartingale is an ((1it)t> O, P) semimartingale also. We also record a trivial observation as a second elementary theorem.

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 lHl denote the filtration obtained by expanding IF with the one a-algebra Q (that is, 1it = Ft V Q, each t > 0 and 1it = 1i t+ ) · Then X is an lHl semimartingale with the same decomposition. Proof Since the local martingale M remains a local martingale under lHI, the

Theorem 2.

D

theorem follows.

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 lHl = (1it)t>o to be the smallest filtration satisfying the usual hypotheses, such that Z1 is 1io measurable and Ft C 1it for all t > 0.

The Levy process Z all t > 0, then the process

Theorem 3 (Ito's Theorem for Levy Processes) .

is an lHl semimartingale. If moreover E {I Zt I } < oo, t ilt Zt - Zs d { Mt = Zt }0 1 - s s is an lHl martingale on [0, oo) . Proof We begin by assuming E{ Zl} < oo , each t > 0. Without loss of gener­ ality we can further assume E{ Zt } = 0. Since Z has independent increments, we know Z is an IF martingale. Let 0 < s < t < 1 be rationals with s = j / and t = kjn. We set Yi = Z i+, - Zi_ . Then Z1 - Z8 = E� / Yi and Zt - Zs = E: } }i . The random variables Yi n

n

are i.i.d. and integrable. Therefore

n

2 Initial Expansions

357

k- 1 n-1 . n- 1 E{Zt - Zs i Z1 - Zs} = E{L Yi l L Yi } = � = L Yi = t - s (Z1 - Zs)· 1-s The independence of the increments of Z yields E{Zt - Zs i H s} = E{Zt ­ therefore E{Zt - Zsi H s} = � �(Z1 - Zs) for all rationals, 0 < Zss i Zt1 -< Zs}; IF martingale, the random variables (Zt)o
.

t=}

.

1. . J t=}

.

•=J

.

an

= 0.

1

There is a potential problem at t = because of the possibility of an explo­ sion. Indeed this is typical of initial enlargements. However if we can show I Z�:::;· I ds} < oo this will suffice to rule out explosions. By the sta­ < tionarity and independence of the increments of we have 2 2 < s) 2 for some constant and for all s, 0 < s < 1. IZ�::::;· I ds } < � ds < oo . Note that if t > then Therefore + zi:::;· ds is a martingale. Since = and it follows that = we have that is a semimartingale. Next suppose only that < oo, t > 0 instead of being in L2 as we assumed earlier. We define

E{f01 E{ (Z1 - Z,) } 1 a (1 a J01 E {f0 M :Ft 1-lt , Z E{ I Zt l } 1

Z

1

E{ I Z1 - Zs l } 1 Zt Mt J01 Z

Z a

O< s
(Since has dtdlag paths a.s. for each w there are at most a finite number of jumps bigger than a fixed size on each compact time set; hence each Jj is finite a.s.) By the results of Chap. I on Levy processes we have that yt = Jl + J'f is a Levy process with bounded jumps, and hence it is square integrable. and J2 are jointly independent. Additionally, the processes Combining these facts with the preceding proof yields

Zt-

Y, J1 ,

t Y1 - Ys Nt = yt - 10 1 - s ds martingale, where lHl is the smallest right continuous filtration obis an tained by expanding IF with ( Y1 , Jf , J[). Note that JHI' . The same argu­ ment, plus the observation that since Ji does not jump at time t = 1 and is lHl

'

lHl

C

358

VI Expansion of Filtrations

1, t (Jti - Jrot Jj1 _8- J; ds)t>o 1 ' E{f01 I Jt-'- sJ' I ds}

therefore constant in a (random) neighborhood of which in turn yields that · s a loca1 martmg · ale £or < oo a. s. , shows tha lHl . Moreover it is a martingale for lHl as soon as < oo. But the function t ai t for all i by the stationarity of the increments. Hence

Jfo1' 1 Jj1-_8J; I ds

f-->

E{Jf} =

lHl' J1 , J2 Z lHl Z. Jl = "2:: .6.Zs1{ 1 .6.Zsl > 1 } and also Xt = Zt - Jl. O<s< t Then X is also a Levy process, and since X has bounded jumps it is in LP for all > 1, and in particular E{Xl} < each t > 0. Let lHl(Xl) denote IF expanded by the adjunction of the random variable X1 . Let = lHl(X l ) the filtration generated by IHI (Xl ) and lHl(J[). Then lHl(Z1 ) But lHl(Jf), a semimartingale on lHl(Xl), and since J 1 is independent of X we have Xby isTheorem 2 that X is a (JK, P) semimartingale. Therefore by Stricker's Theorem, X is an (lHl, P) semimartingale; and since Jl is a finite variation process adapted to lHl we have that Z is an (lHl, P) semimartingale as well. D The most important example of the above theorem is that of Brownian motion, which was Ito's original formula. In this case let lHl = lHl(B1 ), and we have as a special case the lHl decomposition of Brownian motion: t B� - Bs t B� - Bs t B� - Bs f f Bt = (Bt - h - s ds) + }0 - s ds = f3t + h - s ds. Note that the martingale f3 in the decomposition is continuous and has [tion. /3, f3]tA=simple t, which by Levy's theorem gives that f3 is also a Brownian mo­ calculation gives Ito's original formula, for a process H which is IF predictable: t t t 1 1 B1 o HsdBs = B1 1o Hsdf3s + B1 o Hs B11 -- sBs ds 1t BlHsd/38 + 1t B1 Hs B11 -- Bs s ds 0t 0 = lo B1 HsdBs and are all independent, we conclude that M is an martin­ Since Y, gale. Since M is adapted to lHl, by Stricker's Theorem it is also an !HI martingale, and thus is an semimartingale. Finally we drop all integrability assumptions on We let

p

oo,

=

lK

C lK.

V

2 Initial Expansions

359

where since the random variable B 1 is 1t0 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 < t 1 < < tn < oo. Let lHl denote the smallest filtration satis­ fying the usual hypotheses containing IF and such that the random variables Zt1 , , Ztn are all 1t0 measurable. Then Z is an lHl semimartingale. If we have a countable sequence 0 = to < i t < < tn < , we let T = supn tn , with lHl the corresponding filtration. Then Z is an lHl 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 oo a.s. such that E{IXrn l {Tn >O} I} < oo for each n. Of course, if Xo = 0 this reduces to the condition E{IXrn f} < oo for each n.

Let M be an IF local martingale and suppose M is a semimartin­ gale in an expanded filtration JHI. Then M is a special semimartingale in JHI. Proof. First recall that any local martingale is a special semimartingale. In particular the process Mt = sups
of Chap. III), and this of course remains locally integrable in the expanded filtration lHI, since stopping times remain stopping times in an expanded fil­ tration. Since M is an lHI semimartingale by hypothesis, it is special because Mt is locally integrable (see Theorem 34 of Chap. III). 0

Theorem 5. Let M be an IF local martingale, and let H be predictable such that J� H'1d[M, M]s is locally integrable. Suppose lHl is an expansion of IF such that M is an lHl semimartingale. Then M is a special semimartingale in lHl and let M = N + A denote its canonical decomposition. The stochastic inte­ gral process (f� H8 dMs)t> O is an lHl semimartingale if and only if the process (f� H8 dA s)t >o exists as a path-by-path Lebesgue-Stieltjes integral a.s. First assume that E{f;' H'1d[M, M] s } < oo, which implies that H · M (where H M denotes the stochastic integral process (f� H8 dMs)t>o) is a square integrable martingale, and hence by Theorem 4 it is a special semi­ martingale in the lHl filtration. Let M = N + A be the canonical lHl de­ composition of M, and it follows that H M = H N + H A is the canonical lHI decomposition of H · M. By the lemma following Theorem 23 of Chap. IV we have E{f;' H,?d[N, N] s } < E{f;' H]d[M, M]s } < oo and E{f;' H'1d[A, A]8} < E{f;" H?d[M, M] s } < oo. This allows us to conclude that H is (1t2, M) integrable, calculated in the lHI filtration.

Proof.

·

·

·

·

360

VI Expansion of Filtrations

To remove the assumption E{f;" H'1d[M, M]s} < oo, 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 ·

Let R be a positive, measurable stochas­ tic process. Suppose for almost all s, Rs is independent of :F8; and the law (or distribution) of R8 is 11 which is independent of s, with 11( {0}) = 0 and I;' x11(dx) < oo. If a is a positive predictable process with I� a8ds < oo a.s. for each t, then the two sets below are equal almost surely: 00 00 { 1 Rs a sds < oo} = { 1 a8ds < oo} a.s. Theorem 6 (Jeulin's Lemma) .

We first show that {f;" R8a,ds < oo} C {f;" a8ds < oo} a.s. Let A be an event with P(A) > 0, and let J = 1A, Jt = E{ J l:Ft } , the cadlag version of the martingale. Let j = inft Jt . Then j > 0 on { J = 1}. We have a.s.

Proof

Consider next by Jensen's inequality, and this is equal to (E{1A l:Ft} - 11(0, u]) + , where 11 is the law of Rt . Continuing with ( *) we have

where

0 on (0, 1] because 11( {0}) = 0. Choose An = {f;" R8a8ds < n} for the event A in the foregoing. Then

00

00

oo > nP (A n ) > E{1An 1 Rsasds} = E{ 1 1An R8a8ds} 00 00 E{1A n Rsl :Fs }a . ds} > E{ 1

and therefore I;' a8ds < oo a.s. on { J = 1 }. That is, I;' a8ds < oo a.s. on An . Letting n tend to oo through the integers gives {I;" R8a8ds < oo} C

2 Initial Expansions

361

{I;' a8ds < oo} a.s. We next show the inverse inclusion. Let Tn = inf{t > 0 : I� a8ds > n} . Then Tn is a stopping time, and n > E{f{n a8 ds}. Moreover Tn Tn E{ }{ Rs asds} = E{ }{ E{RsiFs i\Tn }a8ds} 0 0 Tn Tn = E{ 0 E{E{Rs iFs} I Fs i\Tn }asds} = E{ 0 E{Rs}a8ds} Tn a 8ds} < o:n < oo = o:E{ 0 where is the expectation of Rs , which is finite and constant by hypothesis. Therefore IoTn R8a8ds < oo a.s. Let w be in {I;' a 8 ds < oo }. Then there exists an n ( depending on w ) such that Tn(w) = oo. Therefore we have the inclusion o {f;' Rs a sds < oo} :J {f;' a8ds < oo} a.s. and the proof is complete.

1 1

1

o:

We are now ready to study the Brownian case. The next theorem gives the main result.

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 I; vf=s id[M, B]sl < oo a.s. In this case Mt - I� i\1 Bl=�· d[M, B]s is an JH[ local martingale.

Theorem 7.

Proof.

By the Martingale Representation Theorem we have that every IF lo­ cal martingale M has a representation M = M0 + I� H8dB8 , where H is predictable and I� H'1ds < oo a.s., each t > 0. By Theorem 5 we know that M is an JH[ semimartingale if and only if I; I Hs i i B�::::�'I ds is finite a.s., 0 < t < 1 . We take a s = �' and Rs = 1{s < 1 } IB�,I . Then I01 IHsi i B�::::�· I ds = I01 a 8R8ds, which is finite only if I01 a 8ds < oo a.s. by Jeulin ' s Lemma. Thus it is finite only if I01 ds < oo a.s. But

t

*"

101 )1I Hsl- s ds 10t v11- s I Hsl d[B, B]s 1ot v1 1- s id[H · B, B]si =

=

and this completes the proof.

=

=

10t v11- s d[IH I · B, B] s 1ot v1 1- s jd[M, B]sl ·

1/2 < o: < 1, and define 1 Hs = )1 - s ( - ln ( 1 - s)) - "' 1{.! <s< 1 } ·

As an example, let

2

0

362

VI Expansion of Filtrations

Then H is trivially predictable and also f01 H'1ds < oo . However J01 Hs )=s ds 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(Bl ). 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 Hn is a sequence of simple predictable processes tending uniformly in to zero, then we must have also that the stochastic integrals Hn · X tend to zero in probability. If we expand the filtration by adding a a-algebra generated by is added to a random variable to the IF filtration at time 0 (that is, a 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 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 is an (JE, £ ) -valued random variable, where is a stan­ dard Borel space 1 and £ are its Borel sets, and we let JHI( denote the smallest filtration satisfying the usual hypotheses and containing both and the orig­ inal filtration IF. When there is no possibility of confusion, we will write lHI then can be written in place of lHI(L). Note that if E H� = V where x) f--+ x) is an 0£ measurable function. We next recall two standard theorems from elementary probability theory. =

·

(t ,w)

L

:Fo),

{ L}

L

L

Y(w) G(w, L(w)), =

L) L :Ft a{L}, Y Y :Ft (w, G(w,

E

xn Then Let xn be a sequence of real-valued random variables. converges to 0 in probability if and only if limn -->oo E{ min ( 1 , IXn l)} = 0. A proof of Theorem 8 can be found in textbooks on probability ( see for example [109]). We write 1/\ I X n l for min ( 1 , I X n l ) . Also, given a random variable L, we let Qt (w, dx) denote> the regular conditional distribution of L with respect to :Ft, each t 0. That is, for any A E £ fixed, Qt( · , A) is a version of E{1{ LEA} I:Ft}, and for any fixed w, Qt (w, dx) is a probability on £. A second Theorem 8.

standard elementary result is the following.

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{1 { L E dx } I :Ft}. For a proof of Theorem 9 the reader can see, for example, Breiman [23 , page 79]. 1 (JE, E) is a standard Borel space if there is a set r B, where B are the Borel subsets of R, and an injective mapping ¢ lE r such that ¢ is E measurable and ,p - 1 is B measurable. Note that (Rn, Bn) are standard Borel spaces, 1 < n < Theorem 9.

E

:

->

CXJ .

363 Theorem 10 (Jacod's Criterion) . Let L be a random variable with values in a standard Borel space (lE, £), 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 a -finite measure "'t on (lE, £ ) such that Qt (w, dx) « "'t ( dx) a.s. Then every IF semimartingale X is also an lHl( L) semimartingale. Proof. Without loss of generality we assume Qt (w, dx) « "'t ( 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)ryt(dx). Moreover since E{fJE Qt(·, dx)} = E{E{1 {Y E IE } 1Ft}} = P(Y E JE) = 1, we have 2 Initial Expansions

L

L

E{ Qt( · , dx)} = E{ qt(x, w)ryt(dx)} =

L E{qt(x, w)}'f/t (dx) = 1.

Hence for almost all x ( under "'t ( dx)) , we have qt (x, ·) E L 1 ( dP). Let X be an IF semimartingale, and suppose that X is not an lHl( L) semi­ martingale. Then there must exist a u > 0 and an c: > 0, and a sequence Hn of simple predictable processes for the lHl filtration, tending uniformly to 0 but such that infn E{1 /\ I Hn · X I} > c:. Let us suppose that tn < u, and

n- 1 HJ: = L Jf 1 (t; ,t ;HJ (t) i=O with J[' :Ft , V a {L}. Hence J[' has the form 9 i (w, L(w)), where (w, x) 9i (w, x) is :Ft, 0 £ measurable. Since Hn is tending uniformly to 0, we can take without loss I Hn l < 1/n, and thus we can also assume that l 9i l < 1/n. We write n-1 H� 'x (w) = L 9i (w, x)1 (t, ,t, +1J (t), i =O and therefore (x, w) H� ,x (w) and (x, w) (Hn, x · X)t(w) are each £ 0 :Fu measurable, 0 < t < u. Moreover one has clearly Hn . x = Hn ,L . x. Combining E

f-+

f-+

f-+

the preceding, we have

h h

E{1 /\ I Hn · Xu I} E{ (1 /\ I Hn,x · Xu i)Qu (· , dx)} = E{ (1 /\ I Hn ,x · Xul) qu (·, dx)ryu (dx)} =

=

1 E{(1 1\ I Hn,x · Xu i)qu (·, dx)}ryu (dx)

where we have used Fubini's Theorem to obtain the last equality. However the function hn (x ) = E{(1 /\ IHn ,x · Xu l)qu (·, x)} < E{qu (·, x)} E U(dryu ), and since hn is non-negative, we have

364

VI Expansion of Filtrations n . Xu l} IH lim E{1 /\ n ->oo

r

n . Xu i)qu (·, x)}ryu (dx) IH lim E{(1 1\ n->oo }I. = f lim E{( 1 /\ IH n · Xu l)qu ( ·, x)}ryu (dx) JJE n --H)O =

(*)

by Lebesgue's Dominated Convergence Theorem. However qu (·,x) E L 1 (dP) for a. a. x (under (dryu )), and if we define dR = cqu (·, x)dP to be another proba­ bility, then convergence in ?-probability implies convergence in R-probability, · Xu !)} = 0 as well, which im­ since R « P. Therefore lim n- > oo ER{ (1 /\ plies

I Hn , x 1 . -ER{(l l\ i Hn, x Xu l)} Ep{(1 1\ I Hn, x · Xui )qu (·, x)}

0 = hm ---+ oo C

·

n

=

qu (-, x) E L 1 (dP). Therefore the limit x (under (dTJu )), and we conclude n · Xu !)} = 0, IH lim E{(1 /\ n ->oo

for a.a. x (under (dTJu )) such that the integrand in (*) is zero for a.a.

of

which is a contradiction. Hence X must be a semimartingale for the filtration lHl0 , where Hf = :Ft V a {L}. Let X = M + A be a decomposition of X under lHl0 . Since lHl0 need not be right continuous, t�e local martingale M need not be right _continuous. However if we d�fine Mt = Mt if t is rational; and limu"-,t u EIQ Mu if t is not rational; then Mt is a right continuous martingale for the -filtration lHl where Ht = n u >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.

,

0

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 'T]t on (lE, £) such that Qt ( w, dx) « 'T]t ( dx) a. s. if and only if there exists positive a-finite measure ry(dx) such that Qt(w, dx) « ry(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 > 0 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) �----* qt(x, w) be £ 0 :Ft measurable such that Qt(w, dx) = qt(x, w)TJt(dx). Let a t(x) = E{qt(x, w)}, and define

one

(rt ) {q�(x(�w)) ' X, W

=

0,

t

if at(x) > O, otherwise.

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)T/t(dx) is also a version of Qt(W, dx). Let T/ be the law of L. Then for every positive £ measurable function g we 2 Initial Expansions

have

J g(x)ry(dx) = E{g(L)} = E{ k g(x)Qt(·, dx) } = E{ k g(x)qt(x, ·)r]t(dx)} = k g(x)E{qt(x, ·)}'f/t(dx) = 1 g(x)at(x)ryt(dx)

from which we conclude that at(X)T/t(dx) = rt(w, x)ry(dx), and the theorem is proved.

ry(dx).

Hence,

Qt (w, dx)

= 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 JHI ( L) semimartingale.

Since L is independent of Ft, E{g(L) I Ft} = E{g(L)} for any bounded, Borel function g. Therefore

Proof

E{g( L) IFt} =

1 Qt(w, dx)g(x) = k ry(dx)g(x) = E{g(L)},

from which we deduce Qt(W, dx) = ry(dx), and in particular Qt(w, dx) a.s. , and the result follows from Theorem 11.

«

ry(dx)

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 JHI ( L ) semimartingale.

Proof Let L take on the values . . . . The distribution of L is given by ry(dx) = 2:::: � P( L = o:i )Ea , (dx), where Ea , (dx) denotes the point mass at o: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 1

o: 1 , 0:2 , 0:3 ,

given by

The result now follows from Theorem

11.

0

366

VI Expansion of Filtrations

Corollary 3 (Jacod's Countable Expansion) . Let A = (A1 , A 2 , . . . ) be a sequence of events such that Ai n Aj = 0, i f. j, all in F, and such that U� 1 Ai = D. Let lHl be the filtration generated by IF and A, and satisfying the usual hypotheses. Then every IF semimartingale is an lHl semimartingale. Proof. Define L = I:� 1 2 - i 1A ; . Then lHl = the preceding corollary.

JHI(L) and we need only to apply D

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{B1 }. We let lHl denote JHI(Bl). We have

j

E{g(B 1 )I:Ft} = E{g(B1 - Bt + Bt) I:Ft} = g(x + Bt)"lt(dx) where "lt(dx) is the law of B1 - Bt and where we have used that B 1 - Bt is in­ dependent of Ft. Note that "lt(dx) is a Gaussian distribution with mean 0 and variance (1 -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(Bl ) I F1 } = g(Bl ), which yields Q 1 (w, dx) = E{ B1 (w ) } (dx), which is a.s. singular with respect to Lebesgue mea­ sure. We conclude that any IF semimartingale is also an lHI(B 1 ) 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 = J;' g(s)dB8, where J000 g(s)2ds < oo , g a deterministic function. Let a = inf {t > 0 : ft00 g(s) 2 ds = 0}. If h is bounded Borel, then as in the previous example

l g(s)dBs + 1= g(s)dBs)I:Ft} t J h(l g(s)dBs + x) 'f/t (dx),

E{h(V) I :Ft} = E{h( =

t

where "'t is the law of the Gaussian random variable ft00 g(s)dB8• If a = oo , then "'t is non-degenerate for each t, and "'t 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

Qt (w, h) = E{h(V) IFt} =

2 Initial Expansions

t

J h(fo g(s)dBs + x)TJt(dx) .

367

Hence by Theorem 10 we conclude that every lF semimartingale is an !Hl(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

t

t

Xt = Xo + lo a(Xs)dBs + lo b(Xs)ds

where a and b are Lipschitz. In addition, assume a and b are chosen so that for h Borel and bounded,

J

E{h(X1 )IFt} = h(x)n(1 - t, Xt, x)dx n(1 - t , u , x) is a deterministic function.2 Thus Qt (w, dx) = n(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 lF by initially adding Xb we have by Theorem 10 that every lF semimartingale is an !Hl( X1 ) semimartingale, for 0 < t < 1.

where

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 o be a bounded measurable process. It can be shown that there exists a unique optional process 0 H, also bounded, such that for any stopping time T one has

E{Hr 1 { T
The process 0 H is called the optional projection of H. We remark that 0 Ht = Ht a.s. for each fixed time t, but the null set depends on t. Therefore were we simply to write as a process ( E { Ht I Ft} )t>o , instead of 0 H, it would not be uniquely determined almost surely. This is why we use the optional projection. The uniqueness follows from Meyer's section theorems, which are not treated in this book, so we ask the reader to accept it on faith. 2 Sufficient conditions are known for this to be true. These conditions involve Xo

having a nice density, and requirements on the differentiability of the coefficients. See, for example, [190] and [193] .

368

VI Expansion of Filtrations

Definition. Let H = (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 PH is called the predictable projection of H. It follows that for the optional projection, for each stopping time have 0Hr = E{HriFr} a.s. on {T < oo}

T we

whereas for the predictable projection we have that

PHr = E{HriFr- } a.s. on {T < oo} for any predictable stopping time T. For fixed times t > 0 we have then of course 0Ht = E{Ht iFt} a.s. , and one may wonder why we don't simply use the "process" (E{Ht!Ft})t>o instead of the more complicated object 0H. 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 lF c
Proof

Let

s < t. Then

and the result follows.

D

The Azema martingale, a projection of Brownian motion onto a subfil­ tration to which it is not adapted, is an example of the above lemma. The projection of an increasing process, however, is a submartingale but not an increasing process. Lemma. Let lF c IG, and let X be a
Proof

s < t. Then Zs = E{XsiFs} > E{E{Xtl 9s }IFs} = E{XtiFs } = E{E{Xt i Ft} I Fs} = E{ZtiFs}, Let

and the result follows.

D

3 Progressive Expansions

369

The limitation of the two preceding lemmas is the need to require integra­ bility of the random variables Xt for each t > 0. We can weaken this condition by a localization procedure. Definition. We say that a 1 of IF stopping times increasing a.s. to oo, and such that the stopped processes X Tn can be written in the form x rn = Mn + An where Mn is a 0, and with

Ao = 0.

Let 1 . We set To = 0 and let xn = x rn -X Tn - 1 , and Nn = M Tn - M Tn - 1 with N° = 0. For each n there are two increasing processes en and nn , each starting at 0, with xn = Nn + en - Dn , and moreover we can choose this decomposition Theorem 12 (Filtration Shrinkage) .

such that the following holds:

E {e� } + E {D�}

+ E {sup I Nt i } < oo , t

and where t < Tn - 1 implies er = Df = Nf = 0, and t > Tn implies er erL = Df - DTL = Ntn - N'L = 0. The integrability condition implies that the IF optional projections of en , Dn, and Nn all exist and have dtdh'tg versions. By the previous two lemmas the optional projection of Nn is an IF martingale, and those of en and nn are IF submartingales. Therefore letting o xn , o Nn , oen, and o Dn denote the respective IF optional projections of xn , Nn , en, and Dn, we have that o xn =0 Nn + o en -0 Dn exists and is a special IF semimartingale. Since Tn - 1 and Tn are IF stopping times, we have that also the IF optional projections o Nn , oen, and o Dn and hence o x n are all null over the stochastic interval [0, Tn - d and constant over (Tn , oo ) . Then L > 1 ° xn is a cadlag n D version of oX = Z and thus Z is a special IF semimartingale. -

3 Progressive Expansions

We consider the case where we add a random variable gradually to a filtration in order to create a minimal expanded filtration allowing it to be a stopping time. Note that if the initial filtration IF = (Ft )t>o is given by Ft = {0, D} for all t, then O given by gt = a{ L 1\ s; s < t} is the smallest expansion of IF making L a stopping time. Note that gt = a{ L 1\ t} as well. Let L be a strictly positive random variable. Let A = { (t, w) : t < L (w)}.

VI Expansion of Filtrations

370

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 {0-} is an extra isolated point added to the non-negative reals [0, oo] and which can be thought of as 0- < 0. We also define F0_ = F0 . The purpose of {0-} is to distinguish between the events {w : A(w) = 0} and {w : A(w) = {0}} , each of which could potentially be added to the expanded filtration. The smallest filtration expanding lF and making the random variable L a stopping time is t 9� . Nev­ ertheless, it turns out that the mathematics is more elegant if we consider expansions slightly more rich than the minimal ones. In order to distinguish progressive expansions from initial ones, we will change the notation for the expanded filtrations. Beginning with our usual filtered probability space sat­ isfying the usual hypotheses (D, F, lF, P)m where of course lF denotes the fil­ tration (F)t>o, and a random variable L which is the end of a random set, we define the expanded filtration to be JFL and is given by

r E FF

<=-=?

{ r E F and 3 rt

E

Ft : r n { L > t} = rt n { L > t}}.

This filtration is easily seen to satisfy the usual hypotheses, and also it makes L into a stopping time. Thus
Lemma. If H is a predictable process for JF L then there exists a process J which is predictable for lF such that H = J on [0, L]. Moreover, if T is any stopping time for JF L then there exists an lF stopping time S such that S 1\ L = T 1\ L a.s.

Let r be an event in FF. Then events of the form (t, oo ) X r form a generating set for P(JFL), the predictable sets for JF L . Let H8 = l{(t , oo) xr} (s) and then take J to be 18 = l ( t,oo)x r, (s) . The first result follows by an appli­ cation of the Monotone Class Theorem. For the stopping time T, note that it suffices to take H = l [o,T] , and let J be the IF predictable process guaranteed by the first half of this lemma, and take S = inf{t : lt = 0}. D

Proof.

We next define a measure J.LL on [0, oo] x D by

for any positive, measurable process J. For such a measure J.LL there exists an increasing process l {t> L } which is null at 0- but which can jump at both 0

3

Progressive Expansions

37 1

and +oo. We will denote A L = (Af)t>o, the (predictable) compensator of 1{t > L } for the filtration JF. Therefore if J is an lF predictable bounded process we have

E{ h 1 { L >O - } } = E{ J{ Js dA�}. [o , oo ]

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) JFL process, then its optional projection ° H onto the filtration lF exists. We define Note that 1{ L >t} is decreasing, hence by the lemma preceding Theorem 12 we have that Z is an lF supermartingale. We next prove a needed technical result. Theorem 13. The set {t : 0 < t < oo, Zt - = 0} is contained in the set (L, oo]

and is negligible for the measure dAL . Proof. Let T(w) = inf{t > 0 : Zt (w ) =

0, or Zt_ (w) = 0 for t > 0 } . Then it is a classic result for supermartingales that for almost all w the function t �----> Zt(w) is null on [T(w), oo] . (This result is often referred to as "a non­ negative supermartingale sticks at zero." ) Thus we can write { Z = 0} as the stochastic interval [T, oo] , and on [0, T) we have Z > 0, z_ > 0. We have E{A�A�} = P(T < L) = E{Zr1 { T t} is carried by the graph of L, written [L] , we have L < T, and hence we have Z > 0, z_ > 0 on [0, L). Next observe that the set {Z- = 0 } is predictable, hence 0 = E{l { z, _ =o}d1{ L >t} } = E{1 { z, _ =o } dAf } and hence { z_ = 0} is negligible for dA L . Note that this further implies that D P(ZL - > 0) = 1, and again that {Z_ = 0} C (L, oo].

We can now give a description of martingales for the filtration JFL , 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.

Let Y be a random variable with E{ I YI} < oo. A right con­ tinuous version of the martingale yt = E{YI .Fl} is given by the formula

Theorem 14.

Moreover the left continuous version y_ is given by

372 Proof.

VI Expansion of Filtrations

oL

Let denote the optional a-algebra on JR+ x D, corresponding to coincides with the trace of 0 on [0, L ) (By the filtration JF L . On [0, L ) , 0 we mean of course the optional a-algebra on IR+ x D corresponding to the coincides with the trace of underlying filtration lF.) Moreover on [L, oo ) , the a-algebra B(IR+) 0F00 on [L, oo ) . The analogous description of p L holds, with [0, L ) replaced by (O, L] , and with [L, oo ) replaced with (L, oo ) . It 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, oo ) separately. The second statement follows since z_ > 0 on (0, L] by Theorem 13. D

oL

.

oL

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

c£1 {£> 0}·

Simplifying assumption to hold for the rest of this paragraph .

We 0 for all lF

assume L avoids all lF stopping times. That is, P( L T) stopping times T. Definition. The martingale ML given by MF Af + Zt is called the fun­ damental L martingale. Note that it is trivial to check that ML is in fact a martingale, since A L is the compensator of 1 - Z. Note also that M!:::, = A�, since Zoo 0. Last, note that it is easy to check that ML is a square integrable martingale. Theorem 15. Let X be a square integrable martingale for the lF filtration. L . Moreover Xt/\ L >o for the Then filtration JF is a semimartingale (Xt/\dt /\ J� L Z�- d (X ML )s is a martingale in the JFL filtration. Proof. Let C be the (non-adapted) increasing process Ct = 1{ t > L } · Since C has only one jump at time L we have E{XL } E{f000 X8 dC8 }. 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 00 00 E{ XL } E{ 1 Xs_ dCs } = E{ 1 X dA�} = E{XooA�} = E{X00 M/:::, } = E{[X, ML ] oo } E{ (X, M£ ) 00 }. (* ) Suppose that H is a predictable process for JFL , and J is a predictable process for lF which vanishes on {Z- = 0} 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 h 1 (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 = j 1 (t oo ) and we obtain H · X00 h(XL - Xt)1{t < L } · In this way we can define stochastic integrals for non-adapted simple processes. We have then E{(H X)oo} = E{(J X ) L }, and using our previous calculation, since J · X =

=

=

=

,

=

=

8_

=

=

=

=

,

·

·

is another square integrable martingale, we get

,

3 Progressive Expansions

373

Since (X, M L) is lF predictable, we can replace H by H i�·LJ because it has the same predictable projection on the support of d(X, ML) . This yields

E{

1oo HsdXs } = E{ 1 0

0

L

Hss d(X, ML)s} ·

z

-

Last if we take the bounded lFL predictable process H to be a stochastic interval [0, T 1\ L] , where T is an JFL stopping time, we obtain E{XT/\ L J:/\ L z!_ d(X, ML)s} = 0, which implies by Theorem 21 of Chap. I that 0 Xr/\ L - J:/\ L z�- d(X, ML)s is a martingale.

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 lF filtration. Then (Xt /\d t > o is a semimartingale for the filtration JF L .

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 lF stopping time remains a stopping time for the JF L filtration, the D corollary follows from Theorem 15. 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 < oo there exists an Ft measurable random variable Lt such that L = Lt on { L < t } . Note that in particular if L is honest then it is F00 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 cadlag adapted processes, and let xt * = SUPs< t x: and x + * = sup s x: . Then L = inf{s : xr = x + *}, is honest because on the set { L < t} one has L = inf{ s X8 = X; *}, which is Ft measurable. Theorem 16. L is an honest time if and only if there exists an optional set A c [O, oo] x D such that L(w) = sup {t < oo : (t, w) E A } . :

VI Expansion of Filt rati ons

374

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 < t}, the random variable L coincides with the end of the set An ( [0, t] x 0), which is Ft measurable. For the converse we suppose L is honest. Let (Lt h >o be an lF adapted process such that L = Lt on { L < t}. Since we can replace Lt with Lt 1\ t we can assume without loss of generality that Lt < t. There is also no loss of generality to assume that Lt is increasing with t since we can further replace Lt with sups < t L8• Last, it is also no loss, now that it is increasing, to take it right continuous. We thus have that the process (Lt)t>o is optional for the filtration lF. Last, L is now the end of the optional set {( t , w) Lt (w) = t}. D ;

:

When L is honest we can give a simple and elegant description of the filtration FL . Theorem 1 7. Let L be an honest time. Define gt = { F : F = (A n { L > t}) U ( B n { L

< t}) for some A, B E Ft } Then o constitutes a filtration satisfying the usual hypotheses. Moreover L is a
where H and K are lF predictable processes. Proof. Let < t and take H E g8 , of the form (A n { L > s})U (B n { L < s}) with A, B E F8 • We will show that H E gt , which shows that the collection
is filtering to the right . 3 Since L is an honest time, there must exist D E Ft such that {L < s} = D n {L < t}. Therefore

H n {L < t} = [ (A n Dc) u (B n D)] n {L < t},

with [(A n Dc)U (B n D)] E Ft . The fact that each gt is a a-algebra, and also that IG is right continuous, we leave to the reader. Note that { L < t} E gt which implies that L is a o be a cadlag process adapted to the IG filtration. Let Ht and Kt be Ft measurable random variables such that Ut = Ht on {L > t} and Ut = Kt on {L

< t}.

Next set Ht = lim infs/'t H8; Kt = lim infs/'t 18 for all t > 0. Suppose that is a rational number. Take t = 0 and we have the equality 3 That is, i f s

<

t, then g. C gt ·

s

3

Progressive Expansions

375

U H1 [o,L] + K1 ( L, oo]· =

One can extend this result to predictable processes by the Monotone Class Theorem. 0

Let X be a square integrable martingale for the lF filtration. Then X is a semimartingale for the L } d(X , M L ) s} d(X, M Xt = {Xt 1 - ZsZs L t 1 tAL 1 d(X, M L ) s - 1 {t> L } 1 - Zs d(X, ML ) s} · +{ Zs L

Theorem 18.

1 1

!, 1

0

0

-

Before beginning the proof of the theorem, we establish a lemma we will need in the proof. It is a small extension of the local behavior of the stochastic integral established in Chap. IV. Lemma (Local behavior of the stochastic integral at random times) . Let X and Y be two semimartingales and let H E L(X) and J E L(Y). Let U and V be two positive random variables with U < V. (U and V are not assumed to be stopping times.) Define

A = {w : Ht(w) = lt(w) and Xt(w) - Xu(w) = yt(w) - Yu(w), for all t E [U(w), V(w))}. Let Wt = H · Xt and Zt = J · yt . Then a.s. on A, Wt(w) - Wu(w) = Zt(w) ­ Zu(w) for all t E [U(w), V(w)). Proof. We know by the Corollary to Theorem 26 of Chap. IV that the con­ clusion of the lemma is true when U and V are stopping times. Let ( u , v ) be rationals with 0 < u < v, and let Auv = {w : U(w) < u < v < V(w)} n A. Then also Wt - Wu = Zt - Zu, all t E [u, v), a.s. on Au v , since u, v are fixed times and hence a fortiori stopping times. Next let

{

A = w : n {U(w) < u < v < V(w), and Wt(w) - Wu(w) = Zt(w) - Zu(w) u , v E
for all

}

t E [u, v) } ·

The intersection in the definition of A is countable, so null sets do not accu­ mulate. Finally let u decrease to U(w) and then v increase to V(w), which 0 gives the result.

376 VI Expansion of Filtrations Proof of Theorem 18. We first observe that without loss of generality we can assume X0 0. Let H be a bounded lF predictable process. We define stochas­ =

tic integrals at the random time L by

where H · X (H · Xt )t>o is the lF 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 H and J are both bounded lF pre­ dictable processes and if also H = J on (L, oo] , then J; HsdXs = J; lsdXs a.s., so the definition is well-defined. Since X is an lF martingale in £2 with Xo = 0, we have E{J000 H8dX8} = 0. Applying the equalities ( *) on page 372 to H X we have =

·

where the second equality uses (**) on page 373. Recalling E{f000 H8dX8} = 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- < l } to obtain

because the predictable projection of l -1._ 1( L , oo) (s) is l{Z.-
Up to this point we have not used the hypothesis that L is an honest time. Now we do. Let us take U to be a simple
4 Time Reversal

377

where H and 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
Thus the process X is a
Then X zs a

semimartingale for the
a local martingale and A is a finite variation process. Since A remains a finite variation process in the larger
In this section we apply the results on initial expansions of filtration to some elementary issues regrading time reversal of semimartingales, stochastic inte­ grals , and stochastic differential equations. Definition. Let Y = ( Yf )o
t y = Y(l - t) - - yl _ , Yo - YI - ,

if t = 0, if 0 < t < 1 , if t = 1 .

Note that time as a superscript denotes reversal. Let IF denote the forward filtration, and correspondingly let lF = (P)o
378

VI Expansion of Filtrations

:Ft = J1 V N, where N are the P:_null sets of F1 , while P = :FJ V N is the analogous backward filtration for B. Our first two theorems are obvious, and

we omit the proofs.

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. Theore� 20.

In what follows let us assume that we are given a forward filtration IF = (:Ft)o
[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 = (t0 , . . . , tk ) denote a partition of [0, t] with t0 = 0, tk = t, 0 < t < 1 , and let

Sr, (H, Y) = HoYo + 2 )Ht,+ 1 - Ht; )( Yf,+, - YfJ . i

Definition. Let H, Y be two cadlag processes. The quadratic covariation of H and Y, denoted [ H, Y], is defined to be lim

n--+<Xl

Srn (H, Y) = [H, Y]t t

when this limit exists in probability as mesh(Tt) is cadlag, adapted, and of finite variation a.s.

----+

0, and is such that [H, YJ

be Let Y be an (IF, G) reversible semimartingale, and let H a cadlag process such that Ht E :Ft, and Ht E g l - t , 0 < t < 1 . Suppose the quadratic variation [H, Y] exists. Then the two processes [H, = Y] and Xt J0t H8_ dY8 are (IF, G) reversible semimartingales. Moreover Theorem 22.

Since Ht E g l - t , 0 < t < 1, we have that Hl - s E g l - ( l - s ) = g s , and thus it is adapted and left continuous, so the stochastic integral J� H1 _ 8 dY 8 is well-defined. We choose and fix t, 0 < t < 1 . Let T be a partition of [1 - t, 1], T = ( s 1 , . . . , s n ), chosen such that b.. H8 , = 0 and b..Y8, = 0 a.s.,

Proof

4 Time Reversal

i = 2, . . .

, n -

379

1 . Note that one can always choose the partition points { si } in

this manner because

s f---> Y, (w ) has only countably many jumps, and hence f01 1 { [flY. [ >O} ds = 0 a.s. But then f01 P(I� Ys l > O)ds = 0, which in turn implies that P(I�Ys l > 0) = 0 for almost all s in [0, 1]. We now define three new processes: since for each

w,

n- 2

AT = H( l-t ) - b..Yt - t + L Hs , (Y, i+ 1 - Ys J + Hsn -1 (Yt - - Y, n - 1 ) i=l nBT = L1 Hs,+1- (Ys i+ 1 - - Ys, - ) i=l n-2 CT = H1 - tb..Y1 - t + L {(Hs , + 1 - HsJ (Ys ,+ 1 - Ys J } -

i= l

Tn be a sequence of partitions of [0, 1 ] with limn __,00 mesh(Tn ) = 0. Then n = [H, Y] t_ - [H, Yh - t + b.. H1 - t�Y1 - t lim CT -->oo n = [H, Y] t_ - [H, Y] ( l - t) ________.. = -[H, Y] t Since Y is (IF, G) reversible by hypothesis, we know that CT is
Let

------

by hypothesis, it is a semimartingale. Since H is cadlag we can approximate the stochastic integral with partial sums, and thus

Combining ATn '

BTn ' and CTn ' we get

380

VI Expansion of Filtrations

n-2 = H1 - t f:. Y1 - t + L Hs i+1 (Ys i+1 - Ys, ) + H 1 - (Y1 - - Ys - 1 ) n n- 1 - L Hsi+1 ( Ys, +1 - - Ys; - ) n-2 = H1 - t f:. Y1 - t + L Hsi+ 1 ( f:.Y;,i+ 1 - f:.Y;,J - f:. H1 (Y1 - -Ysn _J - H1f:.YBn - 1 i= 1

i=l

i= 1

·

Since we chose our partitions Tn with the property that t:. Y;, , = 0 a.s. for each partition point si , the above simplifies to which tends to 0 since s 1 decreases to 1 - t, and Sn - 1 increases to 1 . Therefore

This establishes 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 Y is a
-

Example (reversibility for Levy process integrals) . Let Z be a Levy process on [0, 1], and let lHr = (Ht)o
11 1 - t ::) O" { Z(l -u) - - Z(l - v) - ; 0 < u , v < 1} v O" { Z! }, t we have that Zt = 21 - (1 - t) E 11 1 - , since Zt = Zt - a.s. Therefore Ht E 11 1 - t . Using Theorem 22 we conclude that Xt = J� f ( Z8_ )dZ8 is an (IF, lHr) reversible

semimartingale. Example (reversibility for Brownian integrals) . Let B be a standard Brownian motion on [0, 1 ] . Let Lf denote Brownian local time at the level x and let f.L be a signed measure on R Let IF be the minimal filtration of B satisfying the usual hypotheses, and let (Jfut )o
4

Time Reversal

381

We will show U is an (IF, lHI) reversible semimartingale. The hypotheses on f imply that its primitive F is the difference of two convex functions, and therefore Mt = F ( Bt) - F ( B0 ) is an IF semimartingale. Moreover £1t = F(B 1 - t) - F(B 1 ), and since we already know that B 1 _t is an lHr semimartingale by Theorem 3, by the convexity of F we conclude that M is also an lHr semimartingale. Therefore f� j(B,)dB8 is an (IF, lHI) reversible semi­ martingale as soon as fiR Lfry( 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 , lHI) reversible semimartingale, for any signed measure p,. This will imply that U is an (IF, lHI) 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 = fiR(L�-t L� )p,( da), and

�

=

1 lim0 ( - { 1 [a,a+c) (Bs )ds) L�-t - L� = c---> E 11- t 1 1 = lim ( - 1 [a-B1,a -B1 + c) (Bs - B 1 )ds ) c---> 0 _t 1 1 � { 1 [a-B 1 ,a-B 1 +c) ( B u )du) = clim0 (---> 11- t - -Ata - B, , where Af is the local time at level x of the standard Brownian motion B. Therefore At = - fiR(A� - B, )p,( da) . Since Af is lHr adapted, and since B1 E 11°, we conclude A is lHI adapted, and therefore U is an (IF, lHI) reversible

� E

!

E

_

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 and b are Lipschitz, and moreover, Borel and bounded, a-

E{h(X1) 1 Ft}

=

a-

and b are chosen so that for

j h(x)1r(1 - t, Xt, x)dx

h

1r(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,

where

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 X1 to get JH[ 11t nu >t :Fu V O" {XI}, 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 Xt X(t, w, x) is the unique solution of :

:

Xt = X +

=

=

t

=

t

1 O"(Xs)dBs + 1 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 O" and b the flow ¢ is injective. For 0 < s < t < 1 define the function ¢s , t to be the flow of the equation

It now follows from the uniqueness of solutions that Xt = ¢s , t(X8), and in particular X1 ¢t , I(Xt), and therefore ¢t} (X!) Xt, where ¢i} is of course the inverse function of ¢s ,t · Since the solution Xs , t of equation ( ) is g O"{Bv - Bu ; s < u, v < t} measurable, we have ¢t, l E F1 - t , where P O"{lh 0 < s < t} V N where N are thet null sets of F. Let 71 t nu>t p v O"{XI} and we have B is an lHI (11 )o
=

*

=

=

=

=

=

'

Let B be a standard Brownian motion and let X be the unique solution of the stochastic differential equation t t Xt = Xo + O"(Xs)dBs + b(X8)ds

Theorem 23.

1

1

for 0 < t < 1 , where O" and b are Lipschitz, and moreover, O" and b are chosen so that for h Borel and bounded,

f

E{h(X1 ) I:Ft} = h(x)1r(1 - t, Xt, x)dx

where 1r(1 - t, u, x) is a deterministic function. Let JH[ be given by 11tt nu >t :Fu V O"{ XI}. Then B is an (IF, lHI) reversible semimartingale, and ); x1- t satisfies the following backward stochastic differential equation for 0 < t < 1 . In particular, X is an (IF, lHI) reversible semimartingale.

= =

Bibliographic Notes

383

Proof

We first note that B is an (IF, liir) reversible semimartingale as we saw in the example on page 367. We have that [O"(X), B] = J� O"'(X8)0"(X8)ds, which is clearly of finite variation, since 0"1 is continuous, and thus 0"1 (X) has paths bounded by a random constant on [0, 1 ] . In the discussion preceding this theorem we further established that O"(Xt) E Ji l - t , and of course O"(Xt ) E :Ft. Therefore by Theorem 22 we have

Observe that (O"(X), B] t = J11_t O"'(Xs)O"(Xs)ds. Use the change of variable u = 1 - s in the preceding integral and also in the term fL t b(X. )ds to get �

and the proof is complete.

0

Bibliographic Notes

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 [1 1 4] . 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 1 {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 [ 1 07] and

384

Exercises for Chapter VI

also E. Pardoux [190]. Related results can be found in J. Picard [193] . 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 (n, F, IF, B, P) be a standard Brownian motion. Expand IF by the initial addition of O" { B 1 }. Let M be the local martingale in the formula

Bt =

t !\ l B l - Bs 1 ds. Bo + Mt +

1-s 0 Show that the Brownian motion M = (Mt)o 1 } J1 . N1 (b) E{Jt iJ1 , Ql} = t J1 . (c) E{Jt iJd = tJ1 . Exercise 3. In Exercise 1 assume B0 = x and condition on the event { B 1 y }. Show that B is then a Brownian bridge beginning at x and ending at y. (See Exercise 9 below for more results concerning filtration expansions and Brownian bridges. Brownian bridges are also discussed in Exercise 24 of Chap. II, and on page 299.) Exercise 4. Let (n, :F, IF, Z, P) be a standard Poisson process. Expand IF by the initial addition on O"{ Zl }. Let M be the local martingale in the formula t!\ l Z1 - Zs Zt = Mt + ds. 1-s Show that M = (Mt)o
1 o

martingale for the expanded filtration
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{ Ba E U!Ft} · Show that

�u =

i �udu

where

2 (u x) �u = g( u, t, Bt) and g(u ' t ' x) - -r====;=1==7 exp { 2(a - t) } y!21r(a - t) for t < a. Show also that �u = l {BaEU} for t > a. Exercise 7. ( Continuation of Exercise 6.) Show that yau- = 0 except on the null set { Ba = u} and infer that (�u )o
for t < a.

1 �!

EJg (u, s, x) = EJx y!21r(a - t) U - X

exp {

-(u - x) 2 } 2(a - t)

Exercise 9. ( Continuation of Exercises 6, 7, and 8.) Show that one can define a probability Pu on C([O, a], JR) such that Pu (Ba = u) = 1 with Pu absolutely continuous with respect to P on Ft for t < a, and singular with respect to P on Fa· (Hint: Sketch of procedure. Let Da = C([O, a), lR), be the space of continuous functions mapping [0, a] into JR, and define Ml: = �u /�0 on Ft for (of course) t < a. Let Pu denote the ( unique ) probability on Da = C([O , a), JR) given on Ft by dPu = Mtu dP , and confirm that

is a Brownian motion under

Pu . Next show that � ( u, Bt) verifies dA� = g(u, t, Bt) s,

Finally show that

u - Bt a-t

386

Exercises for Chapter VI

Bt -

a-t t Bo - - u = (a - t) a a

1t

1 df3s 0 a-s

under Pu , and thus (a - t) J�(a - s) - 1 d{38 , which is defined only on [0, a ) , has the same distribution under Pu as the Brownian bridge, whence

tlim (a - t) --+ a

1 d(38 = 0 t }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 < oo ) = 1. Let A = (At )t>o be the compensator of 1 {t> T } and let M be the martingale Mt = 1 {t> T } - At. Note that A r A = . Let
M/ = f(Ar )1 {t> T } -

t

lo f(As)dAs

=

f(Ar )1 {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 of the random variable Ar is exponential of parameter 1.

Exercise 11. Let IF , T, A, and
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 Bt I < 1}, 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 =

EX= M{;,.

(b) Show that M L is the only IFL square integrable martingale with this property for all IF square integrable martingales. Exercise 15. Give an example of a filtration IF and a non-negative random variable L and an IF semimartingale X such that the post L process Yt = Xt - Xt /\ L = Xtv L - Xt is not a semimartingale. (Hint: Use the standard Brownian filtrations and let

t

Exercises for Chapter VI

387

1 { { f(s)dBs = �2 j(s)dBs} L = sup{t < 1 : lo lo

where f is the non-random function of Exercise 5.) Exercise 16. Let (n, F, lF, P) be a filtered probability space satisfying the usual hypotheses, let L be a non-negative random variable, let 9oo = Foo V o-{ L}, and let t} ' the IF optional projection of 1 { L >t} ' which is a supermartingale. Recall from Theorem 13 that P(ZL- > 0) = 1. Show that is a
t

t

Xt = Xo + lo o-(Xs - )dZs + lo b(Xs - )ds. Assume that £T and b are smooth, and also assume (which is not always true!) that the flows of X are injective. Find a stochastic differential equation solved by X1- t · Exercise 18. Let (!l, F, lF, B, P) be a standard Brownian space with B, a standard Brownian motion. Let Z denote the random zero set of B, which is a closed, perfect, nowhere dense set. Let { (Ln, Rn ) }n>l denote the random intervals contiguous to Z. Without loss of generality assume that the graphs of Ln , denoted [Ln] , are disjoint random sets. P. A. Meyer's conjecture was that one could expand IF to a filtration o} dYs = Yt, which implies that J� 1 { Y.=o} dY8 = 0, which is a contradiction, since as a consequence of Tanaka's formula J� 1 { Y. =o} dY8 = L� , the Brownian local time at 0.

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deux filtrations. In

Symbol Index

vn

II A I I P

A

A

dx (H, J)

245

fo00 1dAs l

220, 301

101 , 245

£(X)

85 32o

154

ER(z) EH (Z)

322

118

EF-s (Z)

I A I , (IA it) t> O

I A ioo

40 , 1 0 1

(�

·

111

F·A

E{IAirn } < oo

111

IF

40

3, 356

56, 155

IFL

370

bO

102

JFI-'

36

102, 154

Fr

5

.71

16

195

Fr ]]))

]]])n

56, 244 250, 301

Ft

-=
IF

279

355

blL

BMO

280

f)(t, w , x)

(.Ft)t>o

101

E{IAioo} < oo

bP

155

105 293, 309

377

F-S acceptable

279

Symbol Index

404

F-S integral FV

I(M)

82

39, 101

9t

228

(1i2 , X) integrable

163

1 8 1 , 185

Jx

58

jp(N,

A)

L(X)

163

HoA

40

La(X)t

H X

58, 60, 156

Lf

0

Hq

X

245

2 1 2 , 230

212

60, 165, 170

L

He; X

61

L (G)

61

Hg X

1 75

Lucp

57

0

0

0

[H, Y] HP

J; Hs

0

dXs

J;+ HsdXs lHl

1i2

fiP H00

58 , 60 82, 271 60, 156 60, 78

362 357

M2

1 78 , 186

M 2 (A)

182

I I M IIBMO N

195

154

0

102

pi-<

36

1 54 , 244, 193

193 314

p

1 02 , 154, 206

p

293, 309 P

(H1)

302, 328

Q

(H2)

302, 328

Qt(w,

(H3)

329

Ql

lx

1 78

58

J; HsdXs

I

52, 56

(M, N)

244

fa"" H.dX.

lHl(L)

L0

378

I H.dX.

56, 1 0 1 , 102, 1 53, 155, 244

52, 144

rv

JRn

131

dx)

4, 2 14 19

362

Symbol Index

s

51

S(IH!)

s...

53

57

S(o:)

248

S(A)

179

[T]

104

ucp

57

Var(X)

xt+

xc x

x

r

269

10, 1 78

54, 1 72 293, 301 66

[X, X] c [X, Y]

�Xt II X II HP

70, 70, 216, 271

211 211

64, 267 267

y_

64 , 1 59

- t y

377

[Z, Z]

zt

84

•

221

( Z H)t 319 J; (dZs)Hs 319

n

293 , 309

a-slice

248

305

¢

3 0 1 , 382

cp

311

1rn

17

a(L\ L00)

66

[X c , X c]

214

369

ya +

J�

369

[X, X]

70

:

70, 221

X Tn Xf

1 16

373

r-

xn

xAy

373

•

70, 122

xVy

1 1 , 244

x +·

(X, X )

Y""

Varr (X)

X*

244

0

116

X(an )

II X I I H2

{X- < a}

Su cp

265

244

( X c , Xc )

52, 56

rx

II X II sP

70

25, 60, 1 58 245

(( x , w) «

405

131

303

150

Subject Index

a-sliceable

BMO

see

approximation of the compensator by

248 bounded mean oscillation

IF optional projection 369 IF special, G semimartingale F-S

FV

arcsine law 369

150

228

associated jump process

27

s ee

Fisk-Stratonovich

asymptotic martingale (AMART)

see

finite variation process

autonomous

re norm

rfP norm

HP

154 193

converge (pre)locally in norm

SP

228

local time of

230

Banach-Steinhaus Theorem

247

Barlow 's Theorem

260

uniform convergence on

compacts in probability

absolutely continuous 192

absolutely continuous space accessible stopping time

226, 227

bounded functional Lipschitz

BMO BM0

norm

195

space of martingales

bracket process

103

199

97, 299, 384

Brownian motion absolute value of

3

with bounded caglad paths

155 56

217

arcsine law for last exit from zero 228

with cadlag paths

56

Azema's martingale

with caglad paths

56

Brownian bridge

1 23

announcing sequence for stopping time 103

195

66

Brownian bridge

55

space [.. of with caglad paths

angle bracket

257

25

Duality Theorem

191

adapted process decomposable

144

bounded mean oscillation

131

Absolutely Continuous Compensators Theorem

240

Bouleau-Yor formula bounded jumps

9

43

Bichteler-Dellacherie Theorem

244

definition of

definition of

Backwards Convergence Theorem

converge (pre)locally in see

228

260

245

(pre)locally in

norm

204

and last exit from zero 247

218

250, 293

Azema's martingale

(pre)locally in

ucp

Laplacians

see

228 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 caglad 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 F V 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 1 { t> L } for IF 371 compound Poisson process 33 conditional quadratic variation 70, 1 22 Brownian motion 123 polarization identity 123 continuous local martingale part of semimartingale 70, 221 continuous martingale part 191 converge (pre)locally in HP, SP 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

Subject Index Doleans-Dade exponential see stochastic exponential Dominated Convergence Theorem for stochastic integrals in Hr 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 1 1 1 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

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 r.v. 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 355, 369 quasi left continuous 148, 189 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

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

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

Jacod's Countable Expansion 366 Jacod' s Countable Expansion Theorem 53, 356 Jacod's criterion 363 Jacod-Yor Theorem on martingale representation 199 Jensen's Inequality 11 Jeulin's Lemma 360

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 1 1 1 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

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

Subject Index 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, SP 260 local martingale BMO 195 cadlag martingale is 37 compensator of 1 18 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

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, SP 24 7 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, 29 1, 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 £2 projection 9 193 HP norm 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 £ 2 178 Jacod-Yor Theorem 199 local 37

412

Subject Index

Martingale Convergence Theorem 8 martingale representation 203 maximal quadratic inequality 11 measures M2(A) 182 not locally square integrable 126 Optional Sampling Theorem 9 orthogonal 1 79 predictable representation property 182 purely discontinuous part 191 quasimartingale 116 sigma martingale 233 space M2 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 1 1 1 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 1 79 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 37 1 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, SP 260 prelocally in HP, SP 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

Subject Index covariation 66 finite 271 of a semimartingale 66 polarization identity 66, 271 quasi left continuous filtration 148, 189 quasimartingale 1 16 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 (JF, G) reversible 378 (7-t_2, X ) integrable 163 re norm 1 54 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

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 7t2 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 r.v. 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 M2 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 20 1 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 Hr 267 Dominated Convergence Theorem in ucp 174 example that is not local martingale 176 for L 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 L 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 announcmg 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

Subject Index 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

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

r

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 ed. 2003) 52 Robert, Stochastic Networks and Queues (2003) 53 Glasserman, Monte Carlo Methods in Financial Engineering (2003)