Stochastic Processes
This book provides a rigorous yet accessible introduction to the theory of stochastic processes. A significant part of the book is devoted to the classic theory of stochastic processes. In turn, it also presents proofs of well-known results, sometime
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Andrei N. Borodin
Stochastic Processes
Probability and Its Applications Series editors Steffen Dereich, Universität Münster, Münster, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, USA Andreas E. Kyprianou, University of Bath, Bath, UK Sidney I. Resnick, Cornell University, Ithaca, USA Probability and Its Applications is designed for monographs on all aspects of probability theory and stochastic processes, as well as their connections with and applications to other areas such as mathematical statistics and statistical physics. More information about this series at http://www.springer.com/series/4893
Andrei N. Borodin
Stochastic Processes
Andrei N. Borodin St. Petersburg Department of athematical Institute Steklov M St. Petersburg, Russia and Department of Mathematics and Mechanics St. Petersburg State University St. Petersburg, Russia
ISSN 2297-0371 ISSN 2297-0398 (electronic) Probability and Its Applications ISBN 978-3-319-62309-2 ISBN 978-3-319-62310-8 (eBook) https://doi.org/10.1007/978-3-319-62310-8 Library of Congress Control Number: 2017956746 Original Russian edition published by LAN Publishing, St. Petersburg, 2013 © Springer International Publishing AG 2017 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. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This book is published under the trade name Birkhäuser The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
CONTENTS
Preface Notation
Chapter I. Basic facts § 1. § 2. § 3. § 4. § 5. § 6. § 7. § 8. § 9. §10. §11.
Random variables Conditional expectations Stochastic processes. Continuity criterion Stopping times Martingales Markov processes Processes with independent increments Gaussian processes Stationary processes Brownian motion process Brownian bridge
Chapter II. Stochastic calculus § § § § §
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