Singular Stochastic Differential Equations
The authors introduce, in this research monograph on stochastic differential equations, a class of points termed isolated singular points. Stochastic differential equations possessing such points (called singular stochastic differential equations here) ar
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Alexander S. Cherny Hans-J¨urgen Engelbert
Singular Stochastic Differential Equations
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Authors Alexander S. Cherny Department of Probability Theory Faculty of Mechanics and Mathematics Moscow State University Leninskie Gory 119992, Moscow Russia e-mail: [email protected] Hans-J¨urgen Engelbert Institut f¨ur Stochastik Fakult¨at f¨ur Mathematik und Informatik Friedrich-Schiller-Universit¨at Jena Ernst-Abbe-Platz 1-4 07743 Jena Germany e-mail: [email protected]
Library of Congress Control Number: 2004115716
Mathematics Subject Classification (2000): 60-02, 60G17, 60H10, 60J25, 60J60 ISSN 0075-8434 ISBN 3-540-24007-1 Springer Berlin Heidelberg New York DOI: 10.1007/b104187 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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 for prosecution under the German Copyright Law. Springer is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2005 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 specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors 41/3142/ du - 543210 - Printed on acid-free paper
Preface
We consider one-dimensional homogeneous stochastic differential equations of the form dXt = b(Xt )dt + σ(Xt )dBt , X0 = x0 , (∗) where b and σ are supposed to be measurable functions and σ = 0. There is a rich theory studying the existence and the uniqueness of solutions of these (and more general) stochastic differential equations. For equations of the form (∗), one of the best sufficient conditions is that the function (1 + |b|)/σ 2 should be locally integrable on the real line. However, both in theory and in practice one often comes across equations that do not satisfy this condition. The use of such equations is necessary, in particular, if we want a solution to be positive. In this monograph, these equations are called singular stochastic differential equations. A typical example of such an equation is the stochastic differential equation for a geometric Brownian motion. A point d ∈ R, at which the function (1 + |b|)/σ 2 is not locally integrable, is called in this monograph a singular point. We explain why these points are indeed “singular”. For the isolated singular points, we perform a complete qualitative classification. According to this classification, an isolated singular point can have one of 48 possible types. The t
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