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Jean-François Le Gall

Brownian Motion, Martingales, and Stochastic Calculus

Graduate Texts in Mathematics

274

Graduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA

Advisory Board: Alejandro Adem, University of British Columbia David Eisenbud, University of California, Berkeley & MSRI Irene M. Gamba, The University of Texas at Austin J.F. Jardine, University of Western Ontario Jeffrey C. Lagarias, University of Michigan Ken Ono, Emory University Jeremy Quastel, University of Toronto Fadil Santosa, University of Minnesota Barry Simon, California Institute of Technology

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.

More information about this series at http://www.springer.com/series/136

Jean-François Le Gall

Brownian Motion, Martingales, and Stochastic Calculus

123

Jean-François Le Gall Département de Mathématiques Université Paris-Sud Orsay Cedex, France

ISSN 0072-5285 Graduate Texts in Mathematics ISBN 978-3-319-31088-6 DOI 10.1007/978-3-319-31089-3

ISSN 2197-5612 (electronic) ISBN 978-3-319-31089-3 (eBook)

Library of Congress Control Number: 2016938909 Mathematics Subject Classification (2010): 60H05, 60G44, 60J65, 60H10, 60J55, 60J25 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

Preface

This book originates from lecture notes for an introductory course on stochastic calculus taught as part