Seminar on Stochastic Analysis, Random Fields and Applications: Centro Stefano Franscini, Ascona, 1993 PDF
Preview Seminar on Stochastic Analysis, Random Fields and Applications: Centro Stefano Franscini, Ascona, 1993
Progress in Probability Volume 36 Series Editors Thomas Liggett Charles Newman Loren Pitt Seminar on Stochastic Analysis, Random Fields and Applications Centro Stefano Franscini, Ascona, 1993 Erwin Bolthausen Marco Dozzi Francesco Russo Editors Springer Basel AG Editors: Erwin Bolthausen Francesco Russo Institut fiir Mathematik Universite Paris-Nord Abteilung Angewandte Mathematik Departement de Mathematiques Universităt Ziirich-Irchel Institut Galilee Winterthurerstr. 190 Av. Jean-Baptiste Clement CH-8057 Ziirich F-93430 Villetaneuse and Universităt Bielefeld MarcoDozzi BIBOS Departement de Mathematiques D-33615 Bielefeld 1 Universite de Nancy 1 B.P. 239 F-54506 Vandoeuvre-Les-Nancy Cedex A CIP catalogue record for this book is available from the Library of Congress, Washington D. c., USA Deutsche Bibliothek Cataloging-in-Publication Data Seminar on Stochastic AnaIysis, Random Fields and Applications <1993, Ascona>: Seminar on Stochastic Analysis, Random Fields and Applications : Cent ro Stefano Franscini, Ascona, 1993/ Erwin Bolthausen ... ed. - 1995 Springer Basel AG (Progress in probability ; VoI. 36) ISBN 978-3-0348-7028-3 ISBN 978-3-0348-7026-9 (eBook) DOI 10.1007/978-3-0348-7026-9 NE: Bolthausen, Erwin [Hrsg.); GT 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, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 1995 Springer Basel AG Originally published by Birkhăuser Verlag Basel in 1995 Softcover reprint of the hardcover 1s t edition 1995 Printed an acid-free paper produced trom chlorine-free pulp 00 ISBN 978-3-0348-7028-3 987654321 CONTENTS Preface ........................................................ vii List of Participants ............................................ ix Robert Aebi Propagation of chaos - the inverse problem .................. 1 Sergio Albeverio, Yu. G. Kondratiev and Michael Rockner A remark on stachastic dynamics on the infinite-dimensional torus ............................ 27 Rene A. Carmona and Jean Pierre Fouque Diffusion-approximation for the advection-diffusion of a passive scalar by a space-time Gaussian velocity field 37 Rene A. Carmona and J. A. Yan A new space of white noise distributions and applications to SPDE's ................................. 51 Hans Crauel and Franco Flandoli Dissipativity of three-dimensional stochastic Navier-Stokes equation ........................... 67 Ana Bela Cruzeiro, Zbigniew Haba and Jean-Claude Zambrini Bernstein diffusions and Euclidean quantum field theory 77 Rosario Delgado and Marta Sanz-Sole A Fubini theorem for generalized Stratonovich integrals 99 Amir Dembo and Ofer Zeitouni Large deviations via parameter dependent change of measure, and an application to the lower tail of Gaussian processes ....................................... 111 Jon Gjerde, H. Holden, B. 0ksendal, J. Ub¢e and T. Zhang An equation modelling transport of a substance in a stochastic medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123 Zbigniew Haba Stochastic representation of unitary quantum evolution 135 Peter Imkeller and Ferenc Weisz Critical dimensions for the existence of self-intersection local times of the Brownian sheet in Rd ..................... 151 Rcmi Leandre and Francesco Russo Density estimates for stochastic partial differential equations ........................................ 169 v Chun Wa Li Almost sure convergence of stochastic differential equations of jump-diffusion type 187 Paul Malliavin Applications and foundations of quasi sure analysis 199 David Nualart and Josep Vives A duality formula on the Poisson space and some applications ....................................... 205 Michael Oberguggenberger Generalized functions and stochastic processes ............... 215 Karl-Theodor Sturm On the geometry defined by Dirichlet forms 231 Marc Yor Random Brownian scaling and some absolute continuity relationships ................... 243 Boguslaw Zegarlinski Recent progress in the hypercontractive semigroups .......... 253 Financial models Carlo Bianchi, Riccardo Cesari and Lorenzo Panattoni Alternative estimators of a diffusion model of the term structure of interest rates. A Monte Carlo comparison 265 Rainer Buckdahn Backward stochastic differential equations. Option hedging under additional cost ........................ 307 Michel Chatelain and Christophe Stricker Componentwise and vector stochastic integration with respect to certain multi-dimensional continuous local martingales ................................ 319 Nigel J. Cutland, P. Ekkehard Kopp and Walter Willinger Stock price returns and the Joseph effect: A fractional version of the Black-Scholes model ... . . . . . . . . . .. 327 Damien Lamberton Critical price for an American option near maturity ......... 353 Giovanni B. Di Masi, Eckhard Platen and W. J. Runggaldier Hedging of options under discrete observation on assets with stochastic volatility ........................... 359 Wolfgang J. Runggaldier and Martin Schweizer Convergence of option values under incompleteness .......... 365 Agnes Tourin and Thaleia Zariphopoulou Portfolio selection with transaction costs .................... 385 VI PREFACE This volume contains the proceedings of a six-day Seminar on Stochastic Analysis, Random Fields and Applications which took place at the Centro Stefano Franscini (Monte Verita) in Ascona, Switzerland, from Monday June 7 to Saturday June 12, 1993. It was financially supported by ETH Zurich and sponsored by Banca della Svizzera Italiana. The seminar focused on four topics, namely stochastic partial differential equations, analysis of Wiener functionals and non-causal stochastic calculus, random media and financial models. The fourth topic was the subject of a special Minisymposium. The purpose of the seminar was to promote the interaction between specialists in these areas and young scientists. We tried to provide an up-to-date picture of current issues and outstanding problems. We hope that these proceedings will convey this picture to a larger audience. Several lecturers had been asked to present a review of their research areas. Two public lectures were given by Prof. Nicole EI Karoui (Paris 6) and Dott. Riccardo Cesari (Banca d'Italia, Bologna). The titles were: - Modeles stochastiques des taux d'interets, - II mercato finanziario in Italia. Istituzioni e strumenti. Let us briefly discuss the state of the art in the topics of the seminar: Stochastic partial differential equations basically include nonlinear partial dif ferential equations perturbed by Gaussian white or coloured noise. In principle, there are two classes of stochastic partial differential equations: those which admit continuous multi-parameter processes as solutions and those which involve non linearities of Schwartz distributions. The first case arises in low space dimension d, say d = 1 for hyperbolic and parabolic stochastic partial differential equations and d :::; 3 in the elliptic case. Problems of existence, uniqueness and related questions have been considered since the '70s. The hyperbolic case constitutes one extension of the classical theory of two-parameter processes, whose index set is the first quar ter of the plane; this field was very fashionable in the '70s. Since the mid '80s the Malliavin calculus has been applied to establish and discuss existence, regularity, positivity and support of the law density. Particular interest has been devoted to support theorems in the case of a stochastic heat equation with nonlinearities of power type; these equations are strongly connected with limit theorems for interacting particle systems. The case where no continuous solutions exist is par tially motivated by mathematical physics and more specifically by quantum field theory. One of the basic problems in this area is stochastic quantization which is a particular heat equation perturbed by white noise and involving nonlinearities of polynomial or trigonometrical type. The polynomial case has recently been treated vii viii PREFACE with the help of Dirichlet form techniques, which have had a big development in the past years. Non-causal stochastic calculus has mainly been developed in the last ten years after the rediscovery of the Skorohod integral. In the framework of stochastic calculus of variations, this integral can be seen as a divergence, that is, the dual operator of the derivative on the Wiener space. From this starting point, a stochas tic calculus has been developed, sometimes also in connection with two-parameter pocesses and manifold-valued processes, and at the same time, a large class of anticipating stochastic differential equations has been discussed. Extensions from the Brownian case to some jump processes (Poisson processes, queuing processes and so on) deserve particular interest. Under the influence of Malliavin, connec tions with quasi sure analysis have been established. There are other approaches, not directly related to the Skorohod calculus, for instance pathwise approaches. Random media is a rapidly developing field with many successes in the last years and many more unsolved problems. Generally speaking, one is interested in the behaviour of some quantity which is usually well understood in a completely homogeneous medium but which one now would like to understand in a disordered one. This quantity may be connected with some second random mechanism, e. g. the diffusive behaviour of a random walk, which of course is well known in a nonrandom medium but is very challenging for some types of random media. Some of the problems are directly connected with stochastic partial differential equations. The presentations at the conference in this field were naturally restricted to a few highlights. The success of mathematically oriented financial models has been possible through a parallel development of mathematics, computing power and new fi nancial tools (basically options). The fundamental problems are pricing and hedg ing. The starting point was of course the famous Black-Scholes formula. Later, many generalizations were studied, for instance involving jump processes. The field has much profited from very recent developments in stochastic analysis, like forward-backward stochastic differential equations in connection with control the ory, models involving fractional Brownian motion, stable processes and so on. On the computational side, different classes of Monte Carlo methods have been exploited for simulation. We would like to thank the Centro Stefano Franscini and especially its director Prof. Konrad Osterwalder for making this conference possible. The Banca della Svizzera Italiana generously sponsored the seminar. We wish especially to thank Dr. Grandi and Dr. Gysi. We are particularly grateful to Ms. Gerda Schacher and Dr. Uwe Schmock for the tremendous work involved in preparing this volume for publishing; concerning the organization of the seminar we are grateful for the help which we had from Ms. Katia Bastianelli, Ms. Aline Cossu and Ms. Schk6lziger. Zurich, March 1995 Erwin Bolthausen Marco Dozzi Francesco Russo LIST OF PARTICIPANTS Robert Aebi. Universitiit Bern. Switzerland Sergio Albeverio. Ruhr-Universitiit Bochum. Germany Renzo G. Avesani. Milano, Italy Paolo Baldi. Universita di Roma 2. Italy Catherine Bandle. Universitiit Basel. Switzerland Andrew D. Barbour. Universitiit Zurich. Switzerland Erwin Bolthausen. Universitiit Zurich. Switzerland Ulriche Bucher. Basel, Switzerland Rainer Buckdahn. Humboldt-Universitiit. Berlin, Germany Rene Carmona. University of California. Irvine, U.S.A. Fabienne Castell. U niversite Paris-Sud. France Riccardo Cesario Banca d'Italia. Bologna, Italy Mireille Chaleyat-Maurel. Universite Paris VI. France Robert Dalang. Ecole poly technique federale de Lausanne. Switzerland Giuseppe Da Prato. Scuola normale superiore. Pisa, Italy Jean-Dominique Deuschel. Technische Universitiit Berlin. Germany Markus Dozzi. Universite de Nancy 1. France Nicole EI Karoui. Universite Paris VI. France Franco Flandoli. Scuola normale superiore. Pisa, Italy Hans Follmer. Universitiit Bonn. Germany Jean-Pierre Fouque. Ecole Polyt echnique. Palaiseau, France Christoph Gallus. Universitiit Erlangen-Nurnberg. Germany Nina Gantert. Technische Universitiit Berlin. Germany Helyette Geman. Essec. Cergy, France Barbara Gentz. Universitiit Zurich. Switzerland Axel Grorud. Universite de Provence. France Zbigniew Haba. University of Wroclaw. Poland Peter Imkeller. Universite de Besan<;on. France Uwe Kuchler. Humboldt-Universitiit. Berlin, Germany Damien Lamberton. Universite de Marne-Ia-Vallee. Noisy-Ie-Grand, France Remi Leandre. Universite de Nancy 1. France C. W. Li. City Polytechnic of Hong Kong. Hong Kong Terry Lyons. Imperial College. London, Great Britain Paul Malliavin. Paris, France Christiane Mathis. Patria Life Insurance. Basel, Switzerland Franco Moriconi. Universita di Perugia. Italy Masao Nagasawa. Universitiit Zurich. Switzerland ix x LIST OF PARTICIPANTS John Noble. University College. Cork, Ireland David Nualart. Universitat de Barcelona. Spain Michael Oberguggenberger. Universitat Innsbruck. Austria Bernt 0ksendal. University of Oslo. Norway Etienne Pardoux. Universite de Provence. France Monique Pontier. Universite d'Orleans. France Jurgen Potthoff. Universitat Mannheim. Germany Nicolas Privault. Universite d'Evry. France Michael Rockner. U niversitat Bonn. Germany Bernard Roynette. Universite de Nancy 1. France Wolfgang Runggaldier. Universita di Padova. Italy Francesco Russo. Universite Paris-Nord. France Marta Sanz-Sole. Universitat de Barcelona. Spain Bruno Scarpellini. Universitat Basel. Switzerland Christian Schwab. Universitat Fribourg. Switzerland Martin Schweizer. Universitat Gottingen. Germany Josep LIuis Sole I Clivilles. Barcelona, Spain Richard Sowers. University of Southern California. Los Angeles, U.S.A. Christophe Stricker. Universite de Besan<;on. France Theo Sturm. Universitat Erlangen-Nurnberg. Germany Alain-Sol Sznitman. ETH. Zurich, Switzerland Anges Tourin. Universite de Paris IX. France Luciano Thbaro. Universita di Trento. Italy Ali SUleyman Ustunel. ENST. Paris, France Walter Willinger. Bellcore. Morristown, NJ, U.S.A. Marc Yor. Universite de Paris VI. France Jerzy Zabczyk. Polish Academy of Sciences. Poland Bijan Z. Zangeneh. Sharif University of Technology. Teheran, Iran Boguslaw Zegarlinski. Imperial College. London, Great Britain Ofer Zeitouni. Technion Haifa. Israel Xialiau Zhang. Societe generale. Paris, France
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