Loeb Measures in Practice: Recent Advances
This expanded version of the 1997 European Mathematical Society Lectures given by the author in Helsinki, begins with a self-contained introduction to nonstandard analysis (NSA) and the construction of Loeb Measures, which are rich measures discovered in
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Subseries: European Mathematical Society Advisers: Fabrizio Catanese, Ragnar Winther
1751
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Nigel J. Cutland
Loeb Measures in Practice:
Recent Advances
Springer
Author Nigel J. Cutland Mathematics Department University of Hull Hull, HU6 7RX, England ' E-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Cutland, Nigel J.: Loeb measures in practice: recent advances I Nigel 1. Cutland. Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris ; Singapore; Tokyo: Springer, 2001 (Lecture notes in mathematics ; 1751) ISBN 3-540-41384-7
Mathematics Subject Classification (2000): 03Hxx, 26E35, 28E05, 76B03, 76M35, 60Hxx ISSN 0075-8434 ISBN 3-540-41384-7 Springer-Verlag Berlin Heidelberg Ne\yYork 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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- Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany
Typesetting: Camera-ready TEX output by the author SPIN: 10759889 41/3142-543210 - Printed on acid-free paper
Preface
This monograph, based on the author’s 1997 EMS Lectures given at the University of Helsinki in May/June 1997, outlines the Loeb measure construction (a way to construct rich measure spaces using Robinson’s nonstandard analysis) and discusses recent applications in stochastic fluid mechanics, stochastic calculus of variations (“Malliavin calculus” and related topics), and mathematical finance theory. The four lectures in Helsinki were designed for a general audience, as is the expanded version presented here. No previous knowledge of either nonstandard analysis or the fields of application is assumed, beyond the general knowledge of the working mathematician. The aim in Chapter 1 is to provide a brief but coherent account of the fundamentals of nonstandard analysis (NSA) and the Loeb construction that is sufficient to make sense of the applications of the later chapters. For each of these we have endeavoured to provide sufficient by way of introduction to the topics concerned to enable even the reader unfamiliar with them to appreciate the basic ideas of the field and then the particular contributions that can be made using NSA and Loeb measures. In fact, one of the major contributions that NSA has made to many fields of application is to aid in understanding of the basic ideas of that field
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