Table Of ContentStochastic Processes and Related Topics
STOCHASTICS MONOGRAPHS
Theory and Applications of Stochastic Processes
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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
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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
