Gushchin - Stochastic Calculus for Quantitative Finance

Alexander A. Gushchin First published 2015 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Press Ltd 27-37 St Georges Road London SW19 4EU UK www.iste.co.uk Elsevier Ltd The Boulevard, Langford Lane Kidlington, Oxford, OX5 1GB UK www.elsevier.com Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

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The symbol indicates the end of the proof. The symbol := means put by definition. R=(,+) = the set of real numbers, R+=[0,+) .

Rd = d -dimensional Euclidean space. Q = the set of rational numbers, Q+=QR+ . N={1,2,3,} = the set of natural numbers. a b = max{ a , b }, a b = min{ a , b } for a,bR . a + = a 0, a = ( a ) 0 for aR . 1B = the indicator function of the set B . 1B = the indicator function of the set B .

E = expectation. E(|G) = conditional expectation with respect to the -algebrs G . FG=(FG) = the smallest -algebra containing the -algebras F and G . AF=(AF) = the smallest -algebra containing the -algebras F , A .

The arbitrage theory for general models of financial markets in continuous time is based on the heavy use of the theory of martingales and stochastic integration (see the monograph by Delbaen and Schchermayer []. The last section is devoted to a-martingales and the AnselStricker theorem.

Some results are given without proofs. These include the section theorem, classical Doobs theorems on martingales, the BurkholderDavisGundy inequality and Its formula. Our method of presentation may be considered as old-fashioned, compared to, for example, the monograph by Protter [], which begins with an introduction of the notion of a semimartingale; in our book, semimartingales appear only in the final chapter. However, the authors experience based on the graduate courses taught at the Department of Mechanics and Mathematics of Moscow State University, indicates that our approach has some advantages. The text is intended for a reader with a knowledge of measure-theoretic probability and discrete-time martingales. Some information on less standard topics (theorems on monotone classes, uniform integrability, conditional expectation for nonintegrable random variables and functions of bounded variation) can be found in the Appendix.

The basic idea, which the author pursued when writing this book, was to provide an affordable and detailed presentation of the foundations of the theory of stochastic integration, which the reader needs to know before reading more advanced literature on the subject, such as Jacod []. The text is accompanied by more than a hundred exercises. Almost all of them are simple or are supplied with hints. Many exercises extend the text and are used later. The work on this book was partially supported by the International Laboratory of Quantitative Finance, National Research University Higher School of Economics and Russian Federation Government (grant no. 14.A12.31.0007).

I wish to express my sincere thanks to Tatiana Belkina for a significant and invaluable assistance in preparing the manuscript. Alexander Gushchin, Moscow, May 2015