Seminar on Stochastic Analysis, Random Fields and Applications: Centro Stefano Franscini, Ascona, 1993

391 Pages·1995·8.23 MB·English

by  Robert Aebi (auth.), Erwin Bolthausen, Marco Dozzi, Francesco Russo (eds.)

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