Noncausal Stochastic Calculus
This book presents an elementary introduction to the theory of noncausal stochastic calculus that arises as a natural alternative to the standard theory of stochastic calculus founded in 1944 by Professor Kiyoshi Itô. As is generally known, Itô Calculus i
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Noncausal Stochastic Calculus
Noncausal Stochastic Calculus
Shigeyoshi Ogawa
Noncausal Stochastic Calculus
123
Shigeyoshi Ogawa Department of Mathematical Sciences Ritsumeikan University Kusatsu, Shiga Japan
ISBN 978-4-431-56574-1 DOI 10.1007/978-4-431-56576-5
ISBN 978-4-431-56576-5
(eBook)
Library of Congress Control Number: 2017945261 © Springer Japan KK 2017 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, express 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. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Japan KK The registered company address is: Chiyoda First Bldg. East, 3-8-1 Nishi-Kanda, Chiyoda-ku, Tokyo 101-0065, Japan
Preface
The aim of this book is to present an elementary introduction to the theory of noncausal stochastic calculus1 that arises as a natural alternative to the standard theory of stochastic calculus founded by Prof. Kiyoshi Itô. To be more precise, we are going to study in this book a noncausal theory of stochastic calculus based on the noncausal integral that was introduced by the author in 1979. We would like to show not only the necessity of such a theory of noncausal stochastic calculus but also its growing possibility as a tool for modelling and analysis in every domain of mathematical sciences. It was around 1944 that late Prof. Kiyoshi Itô first introduced a stochastic integral (with respect to Brownian motion), called nowadays the Itô integral in his name, and originated the theory of stochastic differential equations. Then not immediately but after the Second World War, the potential importance of his theory on stochastic calculus and stochastic differential equations was recognized by mathematicians worldwide and also by physicists and engineers. They all together welcomed Itô’s theory of stochastic differential equations and joined to develop the theory extensively in various directions following th
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