Stochastic Differential Equations
In previous chapters stochastic differential equations have been mentioned several times in an informal manner. For instance, if M is a continuous local martingale, its exponential ε(M) satisfies the equality $$\mathcal{E}{(M)_t} = 1 + \int_0^t {\mathcal{
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Editors
M. Artin S. S. Chem J. Coates J. M. Fröhlich H. Hironaka F. Hirzebruch L. Hörmander S. Mac Lane C. C. Moore J. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai J. Tits M. Waldschmidt S.Watanabe Managing Editors
M. Herger B. Eckmann S. R. S. Varadhan
Daniel Revuz Mare Yor
Continuous Martingales and Brownian Motion With 8 Figures
Springer-Verlag Berlin Heidelberg GmbH
Daniel Revuz Universite de Paris VII Departement de Mathematiques 2, place de Jussieu F-75251 Paris Cedex 05, France MarcYor Universite Pierre et Marie Curie Laboratoire de Probabilites 4, place de Jussieu, Tour 56 F-75252 Paris Cedex 05, France
Mathematics Subject Classification (1980): 60G07, 60H05
ISBN 978-3-662-21728-3 Library ofCongress Cataloging-in-Publication Data Revuz, D. Continuous Martingales and Brownian motion 1 Daniel Revuz, Mare Y or. p. cm. - (Grundlehren der mathematischen Wissenschaften =A Series of comprehensive studies in mathematics; 293) lncludes bibliographical references (p.) and indexes. ISBN 978-3-662-21728-3 ISBN 978-3-662-21726-9 (eBook) DOI 10.1007/978-3-662-21726-9 1. Martingales (Mathematics) 2. Brownian motion processes. 1. Yor, Mare. Il. Title. III. Series: Grundlehren der mathematischen Wissenschaften; 293. QA276.5.R48 1991 519.2'87 -dc20 90-9812 CIP This work is subject to copyright. AII rights are reserved, whether the whole or part ofthe material is concerned, specifically those oftranslation, reprinting, re-use ofillustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law ofSeptember 9, 1965, in its current version, and a copyright fee must always be paid. Violations fali under the prosecution act of the German Copyright Law. O Springer-Verlag Berlin Heidelberg 1991 Originally published by Springer-Verlag Berlin Heidelberg New York in 1991 Softcover reprint ofthe hardcover lst edition 1991 Typesetting: Asco Typesetting Ltd., HongKong 2141/3140-543210 Printed on acid-free paper
Preface
This book focuses on the probabilistic theory ofBrownian motion. This is a good topic to center a discussion around because Brownian motion is in the intersectioll of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with independent increments; it can actually be defined, up to simple transformations, as the real-valued, centered process with independent increments and continuous paths. It is therefore no surprise that a vast array of techniques may be successfully applied to its study and we, consequently, chose to organize the book in the following way. After a first chapter where Brownian motion is introduced, each of the following ones is devoted to a new technique or notion and to some of its applications to Brownian motion. Among these techniques, two are of paramount importance: stochastic calculus, the use ofwhich pervades the whole book and the p
