CONTEMPORARY MATHEMATICS

336

, APORTACIONES MATEMATICAS SOCIEDAD MATEMATICA MEXICANA

Stochastic Models Seventh Symposium on

Probability and Stochastic Processes June 23-28, 2002

Mexico City, Mexico

Jose M. Gonzalez-Barrios Jorge A. Leon

Ana Meda Editors

Stochastic Models

http://dx.doi.org/10.1090/conm/336

CoNTEMPORARY MATHEMATICS

336

~ APORTACIONES MATEMATICAS ~ SOCIEDAD MATEMATICA MEXICANA

Stochastic Models

Seventh Symposium on Probability and Stochastic Processes

June 23-28, 2002 Mexico City, Mexico

Jose M. Gonzalez-Barrios Jorge A. Le6n

Ana Meda Editors

American Mathematical Society Providence, Rhode Island

Editorial Board of Contemporary Mathematics Dennis DeTurck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

Editorial Board of Aportaciones Matematicas

Luis Gorostiza and Martha Takane, Managing Editors

Jose M. Gonzalez-Barrios Alfredo Nicolas Jose Seade Max Neumann Victor Neumann

Guillermo Pastor Sergio Rajsbaum

Jorge X. Valasco Rafael H. Villarreal

This volume contains the proceedings of the Seventh Symposium on Probability and Stochastic Processes that was held in Mexico City, Mexico, in the Antiguo Colegio de San Ildefonso, National University of Mexico (UNAM), from June 23-28, 2002.

2000 Mathematics Subject Classification. Primary 60E15, 60F10, 60G15, 60G50, 60H05, 60H10, 60H15, 60J60, 91B26, 91B30.

Library of Congress Cataloging-in-Publication Data Symposium on Probability and Stochastic Processes (7th: 2002: Mexico City, Mexico)

Stochastic models: Seventh Symposium on Probability and Stochastic Processes, June 23-28, 2002, Mexico City, Mexico/ Jose M. Gonzalez-Barrios, Jorge A. Le6n, Ana Meda, editors.

p. em. -(Contemporary mathematics, ISSN 0271-4132; 336) (Aportaciones matematicas) Includes bibliographical references. ISBN 0-8218-3466-5 (alk. paper) 1. Stochastic analysis-Congresses. I. Gonzalez-Barrios, Jose M., 1957-. II. Leon, J. A.

(Leon Vazquez, Jorge A.), 1960-. Ill. Meda, Ana, 1965-. IV. Title. V. Series. VI. Contem-porary mathematics (American Mathematical Society) ; v. 336.

QA274.2 .S96 2002 519.212-dc22 2003062763

Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math-ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissioniDams.org.

Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

© 2003 by the American Mathematical Society. All rights reserved. The American MathematicalSociety retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. '

Visit the AMS home page at http: I /www. ams. org/

10 9 8 7 6 5 4 3 2 1 08 07 06 05 04 03

Contents

Preface

Lecture Notes

Stochastic Integration with Respect to Fractional Brownian Motion and Applications

vii

DAVID NUALART 3

Entropy and Economic Equilibrium ESA NUMMELIN 41

Credit Risk- A Survey THORSTEN SCHMIDT AND WINFRIED STUTE 75

Research Papers

Optimal Investment in Incomplete Financial Markets with Stochastic Volatility NETZAHUALCOYOTL CASTANEDA-LEYVA AND DANIEL HERNANDEZ-HERNANDEZ 119

Price Calculation for Power Exponential Jump-Diffusion Models-A Hermite-series Approach MANUEL GALEA, JIN MA, AND SOLEDAD TORRES 137

Conditions for Nonconservativity in Quantum Dynamical Semigroups JULIO C. GARciA AND ROBERTO QUEZADA 161

Some Notes on a Dependency Measure JOSE M. GONZALEZ-BARRIOS 171

An Example of an Averaged Markov Decision Process without Stable Policies JUAN GONZALEZ-HERNANDEZ 181

Closeness Estimates for Sums of Independent Random Variables EVGUENI GORDIENKO, MARIO MENDIETA, AND JUAN RUIZ DE CHAVEZ 185

An Example of Infinite Dimensional Quasi-Helix CHRISTIAN HOUDRE AND JOSE VILLA 195

v

vi CONTENTS

A Non-homogeneous Wave Equation Driven by a Poisson Process JORGE A. LEON AND MONICA SARRA 203

Existence of Self-Intersection Local Time of the Multitype Dawson-Watanabe Superprocess JOSE ALFREDO LOPEZ-MIMBELA AND JOSE VILLA 213

Levy Processes in Banach Spaces: Distributional Properties and Subordination ViCTOR PEREZ-ABREU AND ALFONSO ROCHA-ARTEAGA 225

Phase Space Path Integral Representation for the Solution of a Stochastic Schrodinger Equation LUIS A. RINCON 237

A Note on Covariance Characterization of some Generalized Gaussian Random Fields ANNA TALARCZYK 253

On Two-Parameter Stieltjes Integrals for Functions in Besov-Liouville Spaces and Stochastic Integrals CONSTANTIN TUDOR 259

Preface

The Probability and Stochastic Processes Symposia have been held in different cities of Mexico since 1988, every two years. The main purpose of these events is to promote Probability and Stochastic Processes among researchers and students interested in these and related fields.

The Seventh Symposium on Probability and Stochastic Processes was held in Mexico City in the Antiguo Colegio de San Ildefonso, National University of Mexico (UNAM) from June 23rd to June 28th, 2002.

This volume "Stochastic Models," Proceedings of the Seventh Symposium on Probability and Stochastic Processes, contains the lecture notes of three courses and a collection of refereed original research papers by participants in this event.

Besides the three courses, the Seventh Symposium included five invited talks and several short communications. The courses, which were lectured by David Nualart, Esa Nummelin and Winfried Stute, surveyed topics on fractional Brownian motion, credit risk, economic equilibrium and financial markets, and present some of the most recent advances in these fields. The five invited conferences were given by Ma. Emilia Caballero, Begoiia Fernandez, Jin Ma, Philip Protter and Jaime San Martin.

The research papers in this volume cover topics related to the courses, as well as several other topics of current interest in Probability and Stochastic Processes and their applications.

We are deeply grateful to Karim Anaya, Jose Luis Enriquez and Anabel Lagos for helping us with the editorial work of these proceedings.

Finally we acknowledge the financial support of Conacyt (National Science and Technology Council) through grants 32705-E and 37130-E.

Jose M. Gonzalez-Barrios Jorge A. Leon Ana Meda

vii

The volume includes lecture notes and research papers by participants of the Seventh Symposium on Probability and Stochastic Processes held in Mexico City. The lecture notes introduce recent advances in stochastic calculus with respect to fractional Brownian motion, principles of large deviations and of minimum entropy concerning equilibrium prices in random economic systems, and give a complete and thorough survey of credit risk theory. The research papers cover areas such as financial markets, Gaussian processes, stochastic differential equations, stochastic integration, quantum dynamical semigroups, self-inter-section local times, etc.

Readers should have a basic background in probability theory, stochastic integration, and stochastic differential equations. The book is suitable for graduate students and research mathematicians interested in probability, stochastic processes, and risk theory.

ISBN 0-8218-3466-5

9 780821 834664

CONM/336