Stochastic Analysis in Discrete and Continuous Settings With Normal
This volume gives a unified presentation of stochastic analysis for continuous and discontinuous stochastic processes, in both discrete and continuous time. It is mostly self-contained and accessible to graduate students and researchers having alread
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1982
Nicolas Privault
Stochastic Analysis in Discrete and Continuous Settings With Normal Martingales
123
Nicolas Privault Department of Mathematics City University of Hong Kong Tat Chee Avenue Kowloon Tong Hong Kong P.R. China [email protected]
ISBN: 978-3-642-02379-8 DOI: 10.1007/978-3-642-02380-4
e-ISBN: 978-3-642-02380-4
Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2009929703 Mathematics Subject Classification (2000): 60H07, 60G44, 60G42, 60J65, 60J75, 91B28 c Springer-Verlag Berlin Heidelberg 2009 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: SPi Publisher Services Printed on acid-free paper springer.com
Preface
This monograph is an introduction to some aspects of stochastic analysis in the framework of normal martingales, in both discrete and continuous time. The text is mostly self-contained, except for Section 5.7 that requires some background in geometry, and should be accessible to graduate students and researchers having already received a basic training in probability. Prerequisites are mostly limited to a knowledge of measure theory and probability, namely σ-algebras, expectations, and conditional expectations. A short introduction to stochastic calculus for continuous and jump processes is given in Chapter 2 using normal martingales, whose predictable quadratic variation is the Lebesgue measure. There already exists several books devoted to stochastic analysis for continuous diffusion processes on Gaussian and Wiener spaces, cf. e.g. [51], [63], [65], [72], [83], [84], [92], [128], [134], [143], [146], [147]. The particular feature of this text is to simultaneously consider continuous processes and jump processes in the unified framework of normal martingales. These notes have grown from several versions of graduate courses given in the Master in Imaging and Computation at the University of La Rochelle and in the Master of Mathematics and Applications at the University of Poitiers, as well as from lectures presented at the universities of Ankara, Greifswald, Marne la Vall´ee, Tunis, and Wuhan, at the invitations of G. Wallet, M. Arnaudon, H. K¨ orezlioˇglu, U. Franz, A. Sulem, H. Ouerdiane, and L.M
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