Discretization of Processes
In applications, and especially in mathematical finance, random time-dependent events are often modeled as stochastic processes. Assumptions are made about the structure of such processes, and serious researchers will want to justify those assumptions thr
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Stochastic Modelling and Applied Probability (Formerly: Applications of Mathematics)
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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
Jean Jacod r Philip Protter
Discretization of Processes
Jean Jacod Institut de Mathématiques Université Paris VI – Pierre et Marie Curie Place Jussieu 4 Paris Cedex, Paris 75252 France [email protected]
Managing Editors Boris Rozovski˘i Division of Applied Mathematics Brown University 182 George St Providence, RI 02912 USA [email protected]
Philip Protter Department of Statistics Columbia University Amsterdam Av. 1255 New York, NY 10027 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 978-3-642-24126-0 e-ISBN 978-3-642-24127-7 DOI 10.1007/978-3-642-24127-7 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011941186 Mathematics Subject Classification (2010): 60F05, 60G44, 60H10, 60H35, 60J75, 60G51, 60G57, 60H05, 60J65, 60J25 © 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
Two decades ago the authors of this book undertook the study of the errors one makes when numerically approximating the solutions of stochastic differential equations driven by Lévy processes. In particular we were interested in the normalized asymptotic errors of approximations via an Euler scheme, and it turned out we needed sophisticated laws of large numbers and central limit theorems that did not yet exist. While developing such tools, it became apparent that they would be useful in a wide range of applications. One usually explains the difference between probability and statistics as being that probability theory lays the basis for a family of models, and statistics uses data to infer which member or members of that family best fit the dat
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