Stochastic Integration and Differential Equations

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

Applications of Mathematics Stochastic Modelling and Applied Probability

21

Stochastic Control Stochastic Models in Life Sciences

Edited by Advis ory Board

B. Rozovskii M. Yor D. Dawson D. Geman G. Grimmett

I. K aratzas F. Kelly Y. Le Jan B. 0ksendal E. Pardoux

G. Papanicolaou

Springer Berlin Heidelberg

New York Hong Kong London Milan Paris Tokyo

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

2 Marchuk, Methods 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

Introduction I

.

.

.

.

.

.

.

.

II

.

.

.

.

.

.

.

.

.

.

III

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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

5 6 7 8 9

.

.

.

.

.

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 .

.

.

.

.

.

.

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 .

.

.

.

.

.

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 .

.

.

.

.

.

XIII

Contents

References

.

.

.

Symbol Index Subject Index

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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 = Z