An Informal Introduction to Stochastic Calculus with Applications 9620_9789814678933_tp.indd 1 4/6/15 9:39 am November24,2014 14:49 BC:9370–The(1+1)-NonlinearUniverseoftheParabolicMap Universe pagevi An Informal Introduction to Stochastic Calculus with Applications Ovidiu Calin Eastern Michigan University, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 9620_9789814678933_tp.indd 2 4/6/15 9:39 am Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Calin, Ovidiu. An informal introduction to stochastic calculus with applications / by Ovidiu Calin (Eastern Michigan University, USA). pages cm Includes bibliographical references and index. ISBN 978-9814678933 (hardcover : alk. paper) -- ISBN 978-9814689915 (pbk : alk. paper) 1. Stochastic analysis. 2. Calculus. I. Title. II. Title: Introduction to stochastic calculus with applications. QA274.2.C35 2015 519.2'2--dc23 2015014680 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore RokTing - An Informal Introduction to Stochastic Calculus.indd 1 8/5/2015 3:41:36 PM May15,2015 14:45 BC:9620–AnInformalIntroduction toStochastic Calculus Driver˙book pagev Preface Deterministic Calculus has been proved extremely useful in the last few hun- dred years for describing the dynamics laws for macro-objects, such as plan- ets, projectiles, bullets, etc. However, at the micro-scale, the picture looks completely different, since at this level the classical laws of Newtonian me- chanics cease to function “normally”. Micro-particles behave differently, in the sense that their state cannot be determined accurately as in the case of macro-objects; their position or velocity can be described using probability densities rather than exact deterministic variables. Consequently, the study of nature at the micro-scale level has to be done with the help of a special tool, called Stochastic Calculus. The fact that nature at a small scale has a non-deterministic character makes Stochastic Calculus a usefuland important tool for the study of Quantum Mechanics. In fact, all branches of science involving random functions can be ap- proached by Stochastic Calculus. These include, but they are not limited to, signal processing, noise filtering, stochastic control, optimal stopping, elec- trical circuits, financial markets, molecular chemistry, population evolution, etc. However, alltheseapplications assumeastrongmathematical background, which takes a long time to develop. Stochastic Calculus is not an easy theory to grasp and, in general, requires acquaintance with probability, analysis and measure theory. This fact makes Stochastic Calculus almost always absent from the undergraduate curriculum. However, many other subjects studied at thislevel, suchasbiology, chemistry, economics, orelectrical circuits, mightbe more completely understood if a minimum knowledge of Stochastic Calculus is assumed. The attribute informal, present in the title of the book, refers to the fact that the approach is at an introductory level and not at its maximum math- ematical detail. Many proofs are just sketched, or done “naively” without putting the reader through a theory with all the bells and whistles. The goal of this work is to informally introduce elementary Stochastic Calculus to senior undergraduate students in Mathematics, Economics and Business majors. The author’s goal was to capture as much as possible of the v May15,2015 14:45 BC:9620–AnInformalIntroduction toStochasticCalculus Driver˙book pagevi vi An Informal Introduction to Stochastic Calculus with Applications spirit of elementary Calculus, which the students have already been exposed to in the beginning of their majors. This assumes a presentation that mimics similar properties of deterministic Calculus as much as possible, which facili- tates the understanding of more complicated concepts of Stochastic Calculus. The reader of this text will get the idea that deterministic Calculus is just a particular case of Stochastic Calculus and that Ito’s integral is not a too much harder concept than the Riemannian integral, while solving stochastic differential equations follows relatively similar steps as solving ordinary dif- ferential equations. Moreover, modeling real life phenomena with Stochastic Calculus rather than with deterministic Calculus brings more light, detail and significance to the picture. The book can be used as a text for a one semester course in stochastic calculus and probabilities, or as an accompanying text for courses in other areas such as finance, economics, chemistry, physics, or engineering. Since deterministic Calculus books usually start with a brief presentation of elementary functions, and then continue with limits, and other properties of functions, we employed here a similar approach, starting with elementary stochastic processes, different types of limits and pursuing with properties of stochastic processes. The chapters regarding differentiation and integration followthesamepattern. Forinstance,thereisaproductrule,achain-typerule and an integration by parts in Stochastic Calculus, which are modifications of the well-known rules from elementary Calculus. Inorderto makethebookavailable toawideraudience, wesacrificed rigor and completeness for clarity and simplicity, emphasizing mainly on examples and exercises. Most of the time we assumed maximal regularity conditions for which the computations hold and the statements are valid. Many complicated proofs can be skipped at the first reading without affecting later understand- ing. This will be found attractive by both Business and Economics students, whomight getlostotherwiseinavery profoundmathematical textbook where the forest’s scenery is obscured by the sight of the trees. A flow chart indicat- ing the possible order the reader can follow can be found at the end of this preface. An important feature of this textbook is the large number of solved prob- lemsandexampleswhichwillbenefitboththebeginneraswellastheadvanced student. This book grew from a series of lectures and courses given by the author at Eastern Michigan University (USA), Kuwait University (Kuwait) and Fu- Jen University (Taiwan). The student body was very varied. I had math, statistics, computer science, economics and business majors. At the initial stage, severalstudents readthefirstdraftof thesenotes andprovidedvaluable feedback, supplyinga list of corrections, which is far from exhaustive. Finding any typos or making comments regarding the present material are welcome. May15,2015 14:45 BC:9620–AnInformalIntroduction toStochastic Calculus Driver˙book pagevii Preface vii Heartfelt thanks go to the reviewers who made numerous comments and observations contributing to the quality of this book, and whose time is very much appreciated. Finally, I would like to express my gratitude to the World Scientific Pub- lishing team, especially Rok-Ting Tan and Ying-Oi Chiew for making this endeavor possible. O. Calin Michigan, January 2015 November24,2014 14:49 BC:9370–The(1+1)-NonlinearUniverseoftheParabolicMap Universe pagevi May15,2015 14:45 BC:9620–AnInformalIntroduction toStochasticCalculus Driver˙book pageix List of Notations and Symbols The following notations have been frequently used in the text. (Ω,F,P) Probability space Ω Sample space F σ-field X Random variable X Stochastic process t as−lim X The almost sure limit of X t t t→∞ ms−lim X The mean square limit of X t t t→∞ p−lim X The limit in probability of X t t t→∞ F Filtration t dW N ,W˙ , t White noise t t dt W ,B Brownian motion t t ΔW ,ΔB Jumps of the Brownian motion during time interval Δt t t dW ,dB Infinitesimal jumps of the Brownian motion t t V(X ) Total variation of X t t V(2)(X ),(cid:2)X,X(cid:3) Quadratic variation of X t t t F (x) Probability distribution function of X X p (x) Probability density function of X X p(x,y;t) Transition density function E[·] Expectation operator E[X|G] Conditional expectation of X with respect to G Var(X) Variance of the random variable X Cov(X,Y) Covariance of X and Y ρ(X,Y),Corr(X,Y) Correlation of X and Y A ,FX σ-algebras generated by X X ix