Applications of Stochastic Integral
We assume that \((\Omega ,{\mathcal {F}},P, \{ {\mathcal {F}}_t\}_{t\in [0,\infty )})\) is a standard filtered probability space throughout this section.
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Shigeo Kusuoka
Stochastic Analysis
Monographs in Mathematical Economics Volume 3
Editor-in-Chief Toru Maruyama, Professor Emeritus, Keio University, Tokyo, Japan Series Editors Shigeo Kusuoka, Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan Jean-Michel Grandmont, CREST-CNRS, Malakoff CX, France R. Tyrrell Rockafellar, Dept. Mathematics, University of Washington, Seattle, WA, USA
More information about this series at http://www.springer.com/series/13278
Shigeo Kusuoka
Stochastic Analysis
123
Shigeo Kusuoka Graduate School of Mathematical Sciences The University of Tokyo Tokyo, Japan
ISSN 2364-8279 ISSN 2364-8287 (electronic) Monographs in Mathematical Economics ISBN 978-981-15-8863-1 ISBN 978-981-15-8864-8 (eBook) https://doi.org/10.1007/978-981-15-8864-8 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
To Hiroko
Preface
Martingale theory is now regarded as a foundation of the theory of stochastic processes. It was created essentially by Doob [2]. He refined Kolmogorov’s idea on summation of independent random variables and discovered various kinds of inequalities in martingale theory. He also answered the question concerning in what case one can regard a stochastic process as a continuous process. Martingale theory got connected with stochastic integrals and stochastic differential equations of Itô [5], and made great progress in the 1970s. In particular, the French school established a deep theory including discontinuous martingales. However, the theory of continuous martingales is most important for applications. The term “Stochastic analysis” is often used for the theory of c
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