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Part I Fractional Brownian motion 1 Intrinsic properties of the fractional Brownian motion . . . . . 5 1.1 Fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Stochastic integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Correlation between two increments . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Long-range dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 H¨older continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 Path differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 The fBm is not a semimartingale for H = 1/2 . . . . . . . . . . . . . . . 12 1.9 Invariance principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Part II Stochastic calculus 2 Wiener and divergence-type integrals for fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1 Wiener integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Wiener integrals for H >1/2 . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.2 Wiener integrals for H 1/2 . . . . . . . . . . . . . . . . 39 2.2.2 Divergence-type integral for H 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 Fractional white noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Fractional Girsanov theorem . . . . . . . . . . . . . . . .
Probability and Its applications Anderson: Continuous-Time markov chains Azencott Dacunha-Castelle: Series of Irregular Observations Bass: Diffusions and Elliptic Operators Bass: Probabilistic Techniques in analysis Berglund/Gentz: Noise-Induced Phenomena in Slow-Fast Dynamical Systems Biagini/Hu/oksendal/zhang: Stochastic Calculus for Fractional Brownian Motion and applications Chen: Eigenvalues, Inequalities and ergodic theory Choi. ARMA Model identification Costa/Fragoso/Marques. Discrete-Time Markov Jump Linear Systems Daley/vere-Jones: An Introduction to the Theory of Point Processes Volume 1: Elementary Theory and Methods, Second Edition de la pera/Gine: Decoupling: From Dependence to Independence Durrett: Probability Models for DNA Sequence Evolution Galambos/Simonelli: Bonferroni-type Inequalities with applications Gani(Editor ): The Craft of Probabilistic Modelling Grandell: Aspects of Risk Theory Gut: Stopped Random Walks Guyon: Random Fields on a Network Kallenberg: Foundations of Modern Probability, Second Edition Last brandt. marked Point processes on the real line Leadbetter/Lindgren/Rootzen: Extremes and Related Properties of Random Sequences and processes Molchanov: Theory of random sets Nualart: The Malliavin Calculus and Related Topics Rachev/Riischendorf: Mass Transportation Problems. Volume I: Theory Rachev/Ruschendorf: Mass Transportation Problems. Volume I: Applications Resnick. Extreme Values. Regular variation and Point processes Schmidli: Stochastic Control in Insurance Shedler: Regeneration and Networks of Queues Silvestrov: Limit Theorems for Randomly stopped Stochastic Processes Thorisson: Coupling, Stationarity and Regeneration Todorovic: An Introduction to Stochastic Processes and Their Applications Francesca Biagini Yaozhong Hu Bernt oksendal Tusheng Zhang Stochastic calculus for fractional brownian Motion and applications S ringer Francesca Biagini, PhD Bernt oksendal, PhD Mathematisches institut, LMU Munchen Department of Mathematics Theresienstr. 39 D 80333 University of Oslo, Box 1053 Blindern N-0316 Munich, Germany Oslo and Yaozhong hu. phd Dep. ersity of Kansas, 405 Snow Hall Norwegian School of economics and business Administration(NHH) Lawrence. Kansas 66045-2142 USA Helleveien 30, N-5045, Bergen, Norway Tusheng zhang. PhD Center of Mathematics for Applications(CMA) Department of Mathematics Department of Mathematics University of Manchester, Oxford road University of Oslo, Box 1053 Blindern Manchester M13 9PL N-0316, Oslo, Norway and Center of Mathematics for Applications(CMa) Department of Mathematics University of oslo, Box 1053 Blindern N-0316, Oslo, Norway Series editors P Jagers Stochastic Analysis Group CMA Mathematical statistics Australian National University Chalmers University of Technology Canberra act 0200. Australia SE-412 96 Goteberg, Sweden CC. Heyde TG. Kurtz Stochastic Analysis group, CMA Department of mathematics Australian National University University of wisconsin Canberra act 0200. australia 480 Lincoln drive Madison WI 53706 USA ISBN:978-1-85233-996-8 ISBN:978-1-84628-797-8 DOI:10.1007978-1-84628-797-8 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2008920683 Mathematics Subject Classification(2000): 60G05: 60G07; 60G15: 60H05; 60H10; 60H40: 60H07; 93E20 CSpringer-Verlag London Limited 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permit ted 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 of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers The use of registered names, trademarks, etc, in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed on acid-free paper 987654321 Springer Science+ Business Media SprInger. com To Thilo and to my family FB TO Jun and to ruilong YH To e B 0 To Qingh TZ Preface Fractional Brownian motion(fBm)appears naturally in the modeling of many situations, for example, when describing 1. The widths of consecutive annual rings of a tree 2. The temperature at a specific place as a function of time 3. The level of water in a river as a function of time 4. The characters of solar activity as a function of time 5. The values of the log returns of a stock 6. Financial turbulence, i. e. the empirical volatility of a stock, and other turbulence phenomena 7. The prices of electricity in a liberated electricity market In cases 1 to 5 the corresponding f Bm has Hurst coefficient H> 1/2, which means that the process is persistent. In cases 6 and 7 the corresponding fBm has Hurst coefficient H <1/2, which means that the process is anti-persistent For more information about some of these examples we refer to 209 In addition to the above, it is a mathematically tractable fact that f bm represents a natural one-parameter extension (represented by the hurst pa rameter H) of classical Brownian motion. Therefore, it is natural to ask if a stochastic calculus for f Bm can be developed. This is not obvious since f Bm is not a semimartingale(except when H= 1 /2, which corresponds to the classical Brownian motion case). Moreover, it is not a Markov process ei- ther: so the most useful and efficie al mathematical machineries and techniques for stochastic calculus are not available in the f bm case. There fore, it is necessary to develop these techniques from scratch for the fBm. It turns out that this can be done by exploiting the fact that f Bm is a gaussian process The purpose of this book is to explain this in detail and to give applications of the resulting theory. More precisely, we will investigate the main approaches used to develop a stochastic calculus for f Bm and their relations. We also give some applications, including discussions of the(sometimes controversial)use VIII Preface of f Bm in finance, stochastic partial differential equations, stochastic optimal control and local time for f Bm As shown by the reference section, there is a large literature concerning stochastic calculus for f Bm and its applications. We have tried to cite rigor- ously every paper, preprint, or book we were aware of, and we apologize if we accidentally overlooked some works We want to thank Birgit Beck, Christian Bender, Catriona M. Byrne, Alessandra cretarola Robert Elliott. Nils Christian Framstad Serena Fuschin Thilo Meyer-Brandis, Kirsten Minkos, Sebastian QueiBer, Donna Mary Salopek Agnes Sulem, Esko Valkeila, John van der Hoek, and three anonymous referees for many valuable communications and comments. Yaozhong Hu acknowledges the support of the National Science Foundation under grant No. DMS0204613 and DMSo504783. We are also very grateful to our editors Karen Borthwick Helen Desmond and Stephanie Harding for their patience and support Any remaining errors are ours Francesca Biagini, Yaozhong Hu, Bernt oksendal and Tusheng Zhang Munich. Lawrence. Oslo, and manchester, November 2006 Contents Preface Introduction Part i Fractional brownian motion 1 Intrinsic properties of the fractional Brownian motion 1.1 Fractional brownian motion 1.2 Stochastic integral representation 1.3 Correlation between two increments 55689 1.4 Long-range dependence 1.5 Self-similarity 10 1.6 Holder continuity 11 1.7 Path differentiability 11 1. 8 The fBm is not a semimartingale for H +1/2 12 1.9 Invariance principle 14 Part ii stochastic calculus Wiener and divergence-type integrals for fractional Brownian motion 23 2.1 Wiener integrals 23 2.1.1 Wiener integrals for H>1/2 27 2. 1. 2 Wiener integrals for H< 1/2 34 2.2 Divergence-type integrals for f Bm .37 2.2.1 Divergence-type integral for H> 1/2 2.2.2 Divergence-type integral for H 1/2 41 X Contents 3 Fractional Wick Ito Skorohod(fWIS) integrals for fBm of Hurst index H > 1/2 47 3.1 Fractional white noise 47 3.2 Fractional Girsanov theorem 59 3.3 Fractional stochastic gradient 62 3.4 Fractional wick ito skorohod integral 64 3.5 The -derivative ..65 3.6 Fractional Wick Ito Skorohod integrals in L 68 3.7 An Ito formula 71 3.8 LP estimate for the fWIs integral 75 3.9 Iterated integrals and chaos expansion 78 3.10 Fractional Clark Hausmann Ocone theorem 3.11 Multidimensional fWIS integral 3.12 Relation between the fwis integral and the divergence-ypo..87 integral for H>1/2 4 Wick Ito Skorohod(WIs) integrals for fractional Brownian motion 99 4.1 The M operator 9 4. 2 The Wick Ito Skorohod(WIS)integral 103 4.3 Girsanov theorem 4.4 Differentiation 110 4.5 Relation with the standard malliavin calculus 115 4.6 The multidimensional case 118 5 Pathwise integrals for fractional Brownian motion 123 5.1 Symmetric, forward and backward integrals for fBm 123 5.2 On the link between fractional and stochastic calculus. .......125 5. 3 The case H< 1/2 126 5.4 Relation with the divergence integral .130 5.5 Relation with the fWIs integral 5.6 Relation with the WIs integral 137 6 A useful summary 147 6.1 Integrals with respect to fBm 147 6.1.1 Wiener integrals 147 6.1.2 Divergence-type integrals 150 6.1.3 fWIs integrals 151 6.1.4 WIS integrals 153 6.1.5 Pathwise integrals 154 6.2 Relations among the different definitions of stochastic integral. 155 6.2.1 Relation between Wiener integrals and the divergence. 156 6.2.2 Relation between the divergence and the fWis integral 156 6.2.3 Relation between the fWIs and the WIs integrals 157 6.2.4 Relations with the pathwise integrals 158

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