Stochastic Calculus and Applications

Abstract

York 1982York , 2015 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.Printed on acid-free paper Springer Science+Business Media LLC New York is part of Springer Science+Business Media (www. springer.com) To Juli and Ann I returned, and saw vnder the Sunne, That the race is not to the swift, nor the battell to the strong, neither yet bread to the wise, nor yet riches to men of vnderstanding, nor yet fauour to men of skil; but time and chance happeneth to them all.-Ecclesiastes 9:11 (AV, 1611) Preface to the Second Edition (2015)The theory of probability and stochastic calculus has grown significantly since the publication of the first edition of this book. The theory of stochastic integration and semimartingales, a relatively recent development at the time of the first edition, is now a standard and significant part of the working mathematician's toolkit. Concepts such as Backward SDEs, which were unheard of in 1982 (apart from one paper of Bismut), are now understood to be fundamental to the theory of stochastic control and mathematical finance. Applications of stochastic processes arise particularly in finance and engineering. This book presents a rigorous mathematical framework for these problems in a comprehensive and inclusive way. The general theory of processes was developed in the 1970s by Paul-André Meyer and Claude Dellacherie, but for some years it was little known or appreciated in the 'anglo saxon' world ("sauf que les ingenieurs anglais", as Meyer referred to the remarkable group at Berkeley led by Gene Wong and Pravin Varaiya.). The first edition was an attempt to fill this gap in the English literature.To describe this volume as a second edition is an understatement. The original volume was 300 pages; this has over 650. Consequently, the book contains a large amount of additional material, including much new material.The growth in the discipline over the past 30 years has the consequence that it is even less possible now to attempt to give a comprehen...

Stochastic calculus analysis of optical time-of-flight range imaging and estimation of radial motion

“…Stochastic calculus is concerned with the analysis of noisy processes that vary in time [13,14]. Stochastic calculus is an extension of the familiar deterministic calculus to continuous functions of random variables, where random variables are useful for modelling noise.…”

“…We recall the two types of functions that stochastic calculus operates on [13,14]. The first is the class of deterministic functions that classical calculus operates on.…”

“…If µ and σ do not depend on X then dX is called an Itô process, and we can take the integral directly, if not then some substitution must be made to remove the explicit dependence via Itô's theorem, below. Central to modern stochastic calculus is a class of stochastic processes called martingales [13,14]. A martingale is a stochastic process for which the conditionally expected value of future values of X(t) is the most recently observed one.…”

“…Differentiation in deterministic calculus has an important extension in stochastic calculus known as Itô's theorem [13,14]. which can be intuitively understood as the chain rule for functions of stochastic variables.…”

“…The classical method for estimating instantaneous frequency is to use the Wigner distribution, e.g [11,12], a practical method but one that requires a comparitively large number of measurements to what is typically used in ToF ranging. Stochastic calculus [13,14] has been applied to the estimation of the instantaneous frequency of pure sinusoids [15,16] using statistical bootstrapping, but under restrictive assumptions on the amplitude and the nature of the noise.…”