# Brownian motion calculus

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### Transcript of Brownian motion calculus

- 1. Brownian Motion Calculus Ubbo F Wiersema
- 2. Brownian Motion Calculus
- 3. For other titles in the Wiley Finance series please see www.wiley.com/nance
- 4. Brownian Motion Calculus Ubbo F Wiersema
- 5. Copyright C 2008 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777Email (for orders and customer service enquiries): [email protected] our Home Page on www.wiley.comAll Rights Reserved. No part of this publication may be reproduced, stored in a retrieval systemor transmitted in any form or by any means, electronic, mechanical, photocopying, recording,scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 TottenhamCourt Road, London W1T 4LP, UK, without the permission in writing of the Publisher.Requests to the Publisher should be addressed to the Permissions Department, John Wiley &Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailedto [email protected], or faxed to (+44) 1243 770620.Designations used by companies to distinguish their products are often claimed as trademarks.All brand names and product names used in this book are trade names, service marks, trademarksor registered trademarks of their respective owners. The Publisher is not associated with any productor vendor mentioned in this book.This publication is designed to provide accurate and authoritative information in regard to thesubject matter covered. It is sold on the understanding that the Publisher is not engaged inrendering professional services. If professional advice or other expert assistance is required,the services of a competent professional should be sought.Other Wiley Editorial OfcesJohn Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USAJossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USAWiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, GermanyJohn Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, AustraliaJohn Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, L5R 4J3, CanadaWiley also publishes its books in a variety of electronic formats. Some content that appears in printmay not be available in electronic books.Library of Congress Cataloging-in-Publication DataWiersema, Ubbo F. Brownian motion calculus / Ubbo F Wiersema. p. cm. (Wiley nance series) Includes bibliographical references and index. ISBN 978-0-470-02170-5 (pbk. : alk. paper) 1. FinanceMathematical models. 2. Brownian motion process. I. Title. HG106.W54 2008 332.642701519233dc22 2008007641British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 978-0-470-02170-5 (PB)Typeset in 11/13pt Times by Aptara Inc., New Delhi, IndiaPrinted and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
- 6. Models are, for the most part, caricatures of reality, but if they are good, like good caricatures,they portray, though perhaps in a disturbed manner, some features of the real world. Marc Ka c
- 7. voor Margreet
- 8. ContentsPreface xiii1 Brownian Motion 1 1.1 Origins 1 1.2 Brownian Motion Specication 2 1.3 Use of Brownian Motion in Stock Price Dynamics 4 1.4 Construction of Brownian Motion from a Symmetric Random Walk 6 1.5 Covariance of Brownian Motion 12 1.6 Correlated Brownian Motions 14 1.7 Successive Brownian Motion Increments 16 1.7.1 Numerical Illustration 17 1.8 Features of a Brownian Motion Path 19 1.8.1 Simulation of Brownian Motion Paths 19 1.8.2 Slope of Path 20 1.8.3 Non-Differentiability of Brownian Motion Path 21 1.8.4 Measuring Variability 24 1.9 Exercises 26 1.10 Summary 292 Martingales 31 2.1 Simple Example 31 2.2 Filtration 32 2.3 Conditional Expectation 33 2.3.1 General Properties 34
- 9. viii Contents 2.4 Martingale Description 36 2.4.1 Martingale Construction by Conditioning 36 2.5 Martingale Analysis Steps 37 2.6 Examples of Martingale Analysis 37 2.6.1 Sum of Independent Trials 37 2.6.2 Square of Sum of Independent Trials 38 2.6.3 Product of Independent Identical Trials 39 2.6.4 Random Process B(t) 39 2.6.5 Random Process exp[B(t) 1 t] 2 40 2.6.6 Frequently Used Expressions 40 2.7 Process of Independent Increments 41 2.8 Exercises 42 2.9 Summary 423 It Stochastic Integral o 45 3.1 How a Stochastic Integral Arises 45 3.2 Stochastic Integral for Non-Random Step-Functions 47 3.3 Stochastic Integral for Non-Anticipating Random Step-Functions 49 3.4 Extension to Non-Anticipating General Random Integrands 52 3.5 Properties of an It Stochastic Integral o 57 3.6 Signicance of Integrand Position 59 3.7 It integral of Non-Random Integrand o 61 3.8 Area under a Brownian Motion Path 62 3.9 Exercises 64 3.10 Summary 67 3.11 A Tribute to Kiyosi It o 68 Acknowledgment 724 It Calculus o 73 4.1 Stochastic Differential Notation 73 4.2 Taylor Expansion in Ordinary Calculus 74 4.3 It s Formula as a Set of Rules o 75 4.4 Illustrations of It s Formula o 78 4.4.1 Frequent Expressions for Functions of Two Processes 78 4.4.2 Function of Brownian Motion f [B(t)] 80 4.4.3 Function of Time and Brownian Motion f [t, B(t)] 82
- 10. Contents ix T 4.4.4 Finding an Expression for t=0 B(t) dB(t) 83 4.4.5 Change of Numeraire 84 4.4.6 Deriving an Expectation via an ODE 85 4.5 L vy Characterization of Brownian Motion e 87 4.6 Combinations of Brownian Motions 89 4.7 Multiple Correlated Brownian Motions 92 4.8 Area under a Brownian Motion Path Revisited 95 4.9 Justication of It s Formula o 96 4.10 Exercises 100 4.11 Summary 1015 Stochastic Differential Equations 103 5.1 Structure of a Stochastic Differential Equation 103 5.2 Arithmetic Brownian Motion SDE