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Stochastic Modelling and Applied Probability (Formerly: Applications of Mathematics)

66

Edited by B. Rozovski˘ı P.W. Glynn Advisory Board M. Hairer I. Karatzas F.P. Kelly A. Kyprianou Y. Le Jan B. Øksendal G. Papanicolaou E. Pardoux E. Perkins H.M. Soner

For further volumes: http://www.springer.com/series/602

Rafail Khasminskii

Stochastic Stability of Differential Equations

With contributions by G.N. Milstein and M.B. Nevelson

Completely Revised and Enlarged 2nd Edition

Rafail Khasminskii Mathematics Department 1150 Faculty/Administration Building Wayne State University W. Kirby 656 Detroit, MI 48202 USA [email protected] and Institute for Information Transmission Problems Russian Academy of Scienses Bolshoi Karetny per. 19, Moscow Russia Managing Editors Boris Rozovski˘i Division of Applied Mathematics Brown University 182 George St Providence, RI 02912 USA [email protected]

Peter W. Glynn Institute of Computational and Mathematical Engineering Stanford University Via Ortega 475 Stanford, CA 94305-4042 USA [email protected]

ISSN 0172-4568 Stochastic Modelling and Applied Probability ISBN 978-3-642-23279-4 e-ISBN 978-3-642-23280-0 DOI 10.1007/978-3-642-23280-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011938642 Mathematics Subject Classification (2010): 60-XX, 62Mxx Originally published in Russian, by Nauka, Moscow 1969. 1st English ed. published 1980 under R.Z. Has’minski in the series Mechanics: Analysis by Sijthoff & Noordhoff. © Springer-Verlag Berlin Heidelberg 2012 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. Violations are liable to prosecution under the German Copyright Law. 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: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface to the Second Edition

After the publication of the first edition of this book, stochastic stability of differential equations has become a very popular theme of recent research in mathematics and its applications. It is enough to mention the Lecture Notes in Mathematics, Nos 294, 1186 and 1486, devoted to the stability of stochastic dynamical systems and Lyapunov Exponents, the books of L. Arnold [3], A. Borovkov [35], S. Meyn and R. Tweedie [196], among many others. Nevertheless I think that this book is still useful for