The theory of fractional Brownian motion and other long-memory processes are addressed in this volume. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. Among these are results about Levy characterization of fractional Brownian motion, maximal moment inequalities for Wiener integrals including the values 0<H<1/2 of Hurst index, the conditions of existence and uniqueness of solutions to SDE involving additive Wiener integrals, and of solutions of the mixed Brownian—fractional Brownian SDE. The author develops optimal filtering of mixed models including linear case, and studies financial applications and statistical inference with hypotheses testing and parameter estimation. She proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1929 Yuliya S. Mishura Stochastic Calculus for Fractional Brownian Motion and Related Processes ABC Yuliya S. Mishura Department of Mechanics and Mathematics Kyiv National Taras Shevchenko University 64 Volodymyrska 01033 Kyiv Ukraine
[email protected] ISBN 978-3-540-75872-3 e-ISBN 978-3-540-75873-0 DOI 10.1007/978-3-540-75873-0 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2007939114 Mathematics Subject Classification (2000): 60G15, 60G44, 60G60, 60H05, 60H07, 60H10, 60H40, 91B24, 91B28 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, 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 protective laws and regulations and therefore free for general use. Cover design: design & production GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com Preface For several decades the semimartingale processes were the best model in order to implement many ideas. The stochastic calculus for semimartingales and the general theory of stochastic processes, which are closely connected to the theory of stochastic integration and stochastic differential equations, were originated by N. Wiener (Wie23), P. L´evy (Le48), K. Itˆo (Itˆo42), (Itˆ o44), (Itˆ o51), A.N. Kolmogorov (Kol31), W. Feller (Fel36), J.L. Doob, M. Lo´eve, I. Gikhman and A. Skorohod (the list of related papers and books is very long and we do not mention it here in full). Those ideas were developed further by several authors, among them there are K. Bichteler (Bi81), C.S. Chou, P.A. Meyer and C. Stricker (CMS80), K.L. Chung and R.J. Williams (ChW83), C. Dellacherie (Del72), C. Dellacherie and P.A. Meyer (DM82), C. Dol´eansDade and P.A. Meyer (DDM70), H. F¨ ollmer (Fol81a), P.A. Meyer (Me76) and M. Yor (Yor76). These theoretical data were