Stochastic Processes and Related Topics: Proceedings of the 12th Winter School, Siegmundsburg (Germany), February 27-March 4, 2000 PDF
Preview Stochastic Processes and Related Topics: Proceedings of the 12th Winter School, Siegmundsburg (Germany), February 27-March 4, 2000
Stochastic Processes and Related Topics STOCHASTICS MONOGRAPHS Theory and Applications of Stochastic Processes A series of books edited by Saul Jacka, Department of Statistics, University of Warwick, UK Volume 1 Volume 7 Contiguity and the Statistical Stochastic Processes and Optimal Invariance Principle Control P.E. Greenwood and A.N. Shiryayev Edited by H.J. Englebert, I Karatzas and M. Rockner Volume 2 Malliavin Calculus for Processes Volume 8 with Jumps Stochastic Analysis and Related K. Bichteler, J.B. G raver eaux and J. Topics Jacod Edited by T. Lindstr0m, B. 0ksendal and A.S. Üstünel Volume 3 White Noise Theory of Prediction, Volume 9 Filtering and Smoothing Regularity Theory and Stochastic G. Kallianpur and R.L. Karandikar Flows for Parabolic SPDEs F. Flandoli Volume 4 Structure Selection of Stochastic Volume 10 Dynamic Systems: The Information Stochastic Processes and Related Criterion Approach Topics Sándor M. Veres Edited by H.J. Englebert, H. Follmer and J. Zabczyk Volume 5 Applied Stochastic Analysis Volume 11 Edited by M.H.A. Davis and R.J. Random Probability Measures on Elliott Polish Spaces Hans Crauel Volume 6 Nonlinear Stochastic Integrators, Equations and Flows R.A. Carmona and D. Nualart Stochastic Processes and Related Topics Proceedings of the 12th Winter School Siegmundsburg, Germany 27 February - 4 March 2000 Edited by Rainer Buckdahn Université de Bretagne Occidentale Brest, France Hans-Jürgen Engelbert Friedrich-Schiller- Universitàt Jena, Germany and Marc Yor Université Pierre et Marie Curie Paris, France Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A TAYLOR & FRANCIS BOOK First published 2002 by Taylor & Francis Published 2018 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2002 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Publisher's note This book has been prepared from camera-ready copy supplied by the authors. Every effort has been made to ensure that the advice and information in this book is true and accurate at the time of going to press. However, neither the publisher nor the authors can accept any legal responsibility or liability for any errors or omissions that may be made. In the case of drug administration, any medical procedure or the use of technical equipment mentioned within this book, you are strongly advised to consult the manufacturer's guidelines. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book has been requested. CONTENTS Preface vii Backward Stochastic Differential Equations and Viscosity Solutions of Semilinear Parabolic Deterministic and Stochastic PDE of Second Order 1 Rainer Buckdahn Isolated Singular Points of Stochastic Differential Equations 55 Alexander S. Cherny and Hans-Jiirgen Engelbert On One-Dimensional Stochastic Equations Driven by Symmetric Stable Processes 81 Hans-Jürgen Engelbert and Vladimir P. Kurenok Integral Functionals of Strong Markov Continuous Local Martingales 111 Hans-Jürgen Engelbert and Gunar Tittel On the Approximation of Stochastic Integrals and Weighted BMO 133 Stefan Geiss Minimal Distance Martingale Measures and Optimal Portfolios Consistent with Observed Market Prices 141 Thomas Goll and Ludger Rüschendorf On Generalized z-Diffusions 155 Bronius Grigelionis Portfolio Optimisation with Transaction Costs and Exponential Utility 171 RalfKorn and Silke Laue A Semimartingale Backward Equation Related to the p-Optimal Martingale Measure and the Lower Price of a Contingent Claim 189 Michael Mania, Marina Santacroce and Revaz Tevzadze v VI CONTENTS Subordinators Related to the Exponential Functionals of Brownian Bridges and Explicit Formulae for the Semigroups of Hyperbolic Brownian Motions 213 Hiroyuki Matsumoto, Laurent Nguyen and Marc Yor First Passage Time Structural Models with Interest Rate Risk 237 Marek Rutkowski Pricing Options for Markovian Models 249 Gianmario Tessitore andJerzy Zabczyk Three Intertwined Brownian Topics: Exponential Functionals, Winding Numbers, and Ray-Knight Theorems on Local Time 269 Marc Yor List of Participants 273 Index 277 PREFACE This volume contains a significant part of the contributions held at the 12th Winter School on Stochastic Processes and their Applications, February 27 - March 4, 2000, in Siegmundsburg, a little village in the midst of the Forest of Thuringia, near to the famous Rennsteig, the high-level road over the ridge of hills, sixty miles south-west of Jena (Federal Republic of Germany). The conference was organized by the Friedrich Schiller University of Jena as part of a series of regular meetings in the field of stochas tic processes and their applications which began in 1976. The Winter School was attended by thirty-six participants, among them eighteen from abroad. The scientific programme included ten 90-minutes and twenty 30^5-minutes lectures. The scope of the conference covered aspects of the following topics: • Stochastic analysis, • Mathematical Finance, • Stochastic differential equations and, in particular, • Backward stochastic differential equations, • Stochastic partial differential equations. The Winter School included four series of invited lectures: R. BUCKDAHN (Brest) Backward Stochastic Differential Equations. Semilinear PDE and SPDE, Yu. KABANOV (Besançon) Arbitrage Theory, W.M. SCHMIDT (Frankfurt a.M.) Credit Derivatives and Models for Correlated Defaults, M. YOR (Paris) Three Intertwined Brownian Topics: Exponential Functionals, Winding Numbers, and Ray-Knight Theorems on Local Times. The organizers would like to take the opportunity to thank Rainer Buckdahn, Yuri Ka banov, Wolfgang Schmidt and Marc Yor heartily for their lively and interesting lectures, which had a major influence on the success of the conference. vu VIH PREFACE We hope that the volume will give a representative insight into the work of the Winter School and that it will encourage the worldwide cooperation of probabilists working in the field of stochastic processes and their applications. We would like to thank all speakers and participants for their contribution to the success of the meeting. Particular thanks are due to Jochen Wolf and Gunar Tittel for their par ticipation in the organization of the Winter School and the preparation of this volume. We are also indebted to Nicole Serfling for her valuable assistence in organizing the Winter School. The Winter School was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Graduation College Analytical and Stochastic Structures and Systems. It is a pleasure for us to express our gratitude to the DFG for this financial support without which the conference would not have taken place. Research presented at the Winter School was also partly supported by INTAS-97-30204 which is gratefully acknowl edged. For the Organizers Jena, in March 2001 H.-J. Engelbert BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND VISCOSITY SOLUTIONS OF SEMILINEAR PARABOLIC DETERMINISTIC AND STOCHASTIC PDE OF SECOND ORDER RAINER BUCKDAHN Université de Bretagne Occidentale Département de Mathématiques 6, Av. le Gorgeu, B.R 809, F-29285 Brest Abstract The paper, written in form of a survey but including the proofs of the key results, introduces the reader into the theory o fbackward stochastic differential equations and backward doubly stochastic differential equations. For the Markovian case the connection between these both types of equations and viscosity solutions of deterministic and stochastic parabolic partial differential equations is developed. Key Words Backward stochastic differential equation, backward doubly stochastic differential equation, parabolic PDE, stochastic PDE, Feynman-Kac formula, viscosity solution. 0. INTRODUCTION The aim of this article is, in a first step, to present to the reader an introduction to the theory of backward stochastic differential equations (in short, BSDEs) and then to de velop the theory of BSDEs in a Markovian framework and its connection with viscosity solutions of semilinear second order partial differential equations of parabolic type. Afterwards, this connection is generalized to that between backward doubly stochas tic differential equations (BDSDEs) and stochastic viscosity solutions of semilinear stochastic second order partial differential equations (SPDEs, for short) of parabolic type. 1
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