A d v a n c e d S e r i e s o n S t a t is t ic a l S c i e n c e & V o ! , 2 1 A p p lie d Pnobabillity CHANGE OFTIME AND CHANGE OF MEASURE S e c o n d E d itio n O le E . B am d o rff-N ie lsen A lb e r t Shiryaev World Scien tific C H A N G E OF TIME AND C H A N G E OF MEASURE S e co n d Ed itio n ADVANCED SERIES ON STATISTICAL SCIENCE & APPLIED PROBABILITY Editor: Ole E. Bamdorff-Nielsen Published* Vol. 10 Limit Theorems for Associated Random Fields and Related Systems by A. Bulinski and A. Shashkin Vol. 11 Stochastic Modeling of Electricity and Related Markets by F E. Benth, J. Saltyte Benth and S. Koekebakker Vol. 12 An Elementary Introduction to Stochastic Interest Rate Modeling by N. Privault Vol. 13 Change of Time and Change of Measure by 0. E. Barndorff-Nielsen and A. Shiiyaev Vol. 14 Ruin Probabilities (2nd Edition) by S. Asmussen and H. Albrecher Vol. 15 Hedging Derivatives by T. Rheinlander and J. Sexton Vol. 16 An Elementary Introduction to Stochastic Interest Rate Modeling (2nd Edition) by N. Privault Vol. 17 Modeling and Pricing in Financial Markets for Weather Derivatives \S by F E. Benth and J. Saltyte Benth Vol. 18 Analysis for Diffusion Processes on Riemannian Manifolds by F-Y. Wang Vol. 19 Risk-SensitiveInvestmentManagement by M . H. A. Davis and S. Lleo Vol. 20 Spatial Branching in Random Environments and with Interaction by J . Englander Vol. 21 Change of Time and Change of Measure (2nd Edition) by 0. E. Barndoiff-Nielsen and A. Shiiyaev *To view the complete list of the published volumes in the series, please visit: http://www.worldscientific.com/series/asssap A d v a n c e d S e r i e s o n S t a t is t ic a l S c i e n c e & V o l . 2 1 A p p lie d P r o b a b ilit y C H A N G E O F T IM E A N D C H A N G E O F M E A S U R E Second Edition Ole E. Barndorff-Nielsen Aarhus University, Denmark Albert Shiryaev Steklov Mathematical Institute and Moscow State University, Russia World Scientific NEW J E R S E Y • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 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 Bamdorff-Nielsen, O. E. (Ole E.) Change of time and change of measure / by Ole E Bamdorff-Nielsen (Aarhus University, Denmark), Albert Shiryaev (Steklov Mathematical Institute, Russia & Moscow State University, Russia). — 2nd edition. pages cm. — (Advanced series on statistical science and applied probability ; volume 21) Includes bibliographical references and index. ISBN 978-9814678582 (hardcover : alk. paper) I. Random measures. 2. Stochastic analysis. 3. Stochastic models. 4. Probabilities. 5. Time-series analysis. I. Shiryaev, Albert Nikolaevich. II. Title. QA274.28.B37 2015 519.2'3—dc23 2015009067 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 F o r e w o r d t o t h e S e c o n d E d i t i o n The only change to the First Edition of the present book is th a t an additional Chap. 13 has been added th a t outlines developments in the topics of the book th a t have taken place since the publication of the First Edition. These developments have mainly arisen out of studies of the statistical theory of turbulence, but they encompass also results and applications to financial econometrics. The new material falls within the recently established field termed Ambit Stochastics. Some of the topics not discussed in the original Edition are random measures and Levy bases, metatimes (a multivariate form of timechange), change of Levy measures, and the new classes of processes termed Brownian semistationary (or BSS) processes and Levy semistationary (or LSSD) processes. As in the former part of the book, the concepts of volatility/interm ittency play a central role. V This page intentionally left blank F o r e w o r d The conception of the book, based on LECTURE COURSES delivered by the authors in the last years (Aarhus, Moscow, Barcelona, Halmstad, etc.), is defined in many respects by the desire to state the main ideas and results of the stochastic theory of “change of time and change of measure” . These ideas and results have manifold applications, particularly in Mathematical Finance, Financial Economics, Financial Engineering and Actuarial Busi­ ness, when constructing probabilistic and statistical models adequate to statistical data, when investigating the problems of arbitrage, hedging, ra­ tional (fair) pricing of financial and actuarial instruments, when making decisions minimizing the financial and actuarial risks, etc. The lecture- based character of the book defined as well the style of presentation—we have not aimed to give all and complete proofs, many of which are rather long. Our purpose was different, namely to specify the main, essential top­ ics and results of “change of time and change of measure” , so th a t the readers could make use of them in their theoretical and applied activity. A ck n ow led gm en ts. We express our gratitude to our colleagues, espe­ cially Ernst Eberlein and Neil Shephard, for stimulating discussions. We are grateful to the Thiele Centre (Department of M athematical Sciences, Aarhus University) and the Steklov M athematical Institute (Moscow) for providing excellent opportunities to work on the monograph. The sup­ port of INTAS, RFBR, Manchester University (School of Mathematics), and Moscow State University (Department of Mechanics and M athem at­ ics) is gratefully acknowledged. We thank T .B . Tolozova for her help in preparation of the text for publication. O.E. B.-N., A .N . Sh. ♦ • Vll This page intentionally left blank C o n t e n t s Foreword to the Second Edition v Foreword vii Introduction xiii 1. Random Change of Time I 1.1 Basic D e fin itio n s .................................................................... I 1.2 Some Properties of Change of Time .................................. 4 1.3 Representations in the Weak Sense (X I o T ) , in the Strong Sense (X = X o T ) and the Semi-strong Sense (X a= X o T). I. Constructive E x am p les........................................ 8 1.4 Representations in the Weak Sense ( X I o T ) , Strong Sense (X — X o T) and the Semi-strong Sense (X a= X o T). II. The Case of Continuous Local Martingales and Processes of Bounded V a ria tio n ............................................ 15 2. Integral Representations and Change of Time in Stochastic Integrals 25 2.1 Integral Representations of Local Martingales in the Strong S ense.......................................................................................... 25 2.2 Integral Representations of Local Martingales in a Semi- strong S en se.............................................................................. 33 2.3 Stochastic Integrals Over the Stable Processes and Integral R epresentations....................................................................... 35 2.4 Stochastic Integrals with Respect to Stable Processes and Change of T im e ....................................................................... 38 ix