Barcelona Seminar on Stochastic Analysis: St.Feliu de Guíxols, 1991

246 Pages·1993·16.52 MB·English

by  Paolo Baldi, Marta Sanz-Solé (auth.), David Nualart, Marta Sanz Solé (eds.)

Progress in Probabilty
Volume 32
SeriesEditors
ThomasLiggett
CharlesNewman
LorenPitt
Barcelona Seminar on
Stochastic Analysis
St. Peliu de Guixols, 1991
David Nualart
Marta Sanz Sole
Editors
Springer Basel AG
David Nualart
Marta Sanz Sole
Facultat de Matematiques
Universitat de Barcelona
Gran Via, 585
E-08007 Barcelona
Spain
Library of Congress Cataloging-in-Publication Data
Barcelooa Seminar 00 Stochastic Analysis (1991 : San Feliu de Guixols,
Spain)
Barcelona Seminar on Stochastic Analysis : St. Feliu de Guixols,
1991/ David Nualart, Marta Sanz Sole, editors.
p. cm. - (Progress in probability ; v. 32)
ISBN 978-3-0348-9677-1 ISBN 978-3-0348-8555-3 (eBook)
DOI 10.1007/978-3-0348-8555-3
1. Stochastic analysis-Congresses. 1. Nualart, David, 1951-
II. Sanz Sole, Marta, 1952- III. TItle. IV. Series: Progress
in probability ; 32.
QA274.2.B37 1993
519.2-dc20
Deutsche Bibliothek Cataloging-in-Publication Data
Barcelooa Seminar 00 Stochastic Analysis <1991>:
Barcelona Seminar on Stochastic Analysis : St. Feliu de
Guixols, 1991/ David Nualart ; Marta Sanz Sole, ed. - Basel ;
Boston ; Berlin : Birkhăuser, 1993
(Progress in probability ; VoI. 32)
ISBN 978-3-0348-9677-1
NE: Nualart, David [Hrsg.]; GT
This work is subject to copyright. All rights are reserved, whether the whole or part
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© 1993 Springer Basel AG
Originally published by Birkhlluser Verlag Basel in 1993
Softcover reprint ofthe hardcover Ist edition 1993
Printed on acid-free paper, directly from the authors' camera-ready manuscripts
ISBN 978-3-0348-9677-1
FOREWORD
DuringtheofFall 1991, TheCentredeRecercaMatematica, a research
institute sponsored by the Institut d'Estudis Catalans, devoted a quarter
to the study ofstochastic analysis. Prominent workers in this field visited
the Center from all over the world for periods ranging from a few days to
several weeks.
To take advantage of the presence in Barcelona of so many special
ists in stochastic analysis, we organized a workshop on the subject in Sant
Feliu de Guixols (Girona) that provided an opportunity for them to ex
change information and ideas about their current work. Topics discussed
included: Analysis on the Wiener space, Anticipating Stochastic Calculus
and its Applications, Correlation Inequalities, Stochastic Flows, Reflected
Semimartingales, and others. This volume contains a refereed selection of
contributions from some ofthe participants in this workshop.
We are deeply indebted to the authors ofthe articles for these exposi
tions of their valuable research contributions. We also would like to thank
all the referees for their helpful advice in making the volume a reflection of
the dynamic interchange that characterized the workshop. The success of
the Seminar was due essentially to the enthusiasm and stimulating discus
sions ofall the participants in an informal and pleasant atmosphere. To all
ofthem our warm gratitude.
We extend our thanks to the following institutions: CIRIT (Gener
alitat de Catalunya), DGICYT, Directorate-General for Science Research
and Development (CEE), Institut d'Estadistica de Catalunya, Universitat
de Barcelona. Theireconomicsupport made the Seminar possible. We also
warmly thank the Centre de Recerca Matematica for his valuable collabo
ration and advice during the organization ofthe workshop.
Finally, we would like to express our gratitude to Birkhauser Verlag
for having accepted the volume for publication in Progress in Probability,
thus ensuring wide dissemination ofthis work in stochastic analysis to the
mathematical community.
David Nualart
Marta Sanz Sole
Barcelona, September 1992
List of Participants
A. Alabert. Universitat Autonoma de Barcelona. Spain
P. Baldi. Universita di Tor Vergata. Rom, Italy
G. Benarous. Universite de Paris Sud. France
A. Brandao. Instituto de Fisica e Matematica. Lisboa, Portugal
R. Buckdahn. Humboldt Universitiit. Berlin, Germany
P. Cannarsa. Universita di Tor Vergata. Rom, Italy
P. Cattiaux. Universite de Paris X. France
M. Chaleyat-Maurel. Universite de Paris I. France
C. S. Chou. National Central University. Taipei, Taiwan
A. B. Cruzeiro. Instituto de Fisica e Matematica. Lisboa, Portugal
R. Dalang. Tufts University. U.S.A.
R. Delgado. Universitat Autonoma de Barcelona. Spain
M. Farre. Universitat Autonoma de Barcelona. Spain
M. Ferrante. Universitat de Barcelona. Spain
P. Florchinger. Universite de Metz. France
H. Follmer. Universitiit Bonn. Germany
L.G. Gorostiza. Centro de Investigaci6n y de Estudios Avanzados.
Mexico
A. Grorud. Universite de Provence. France
I. Gyongy. Eotvos University. Budapest, Hungary
P. Imkeller. Ludwig-Maximilians Universitiit. Munchen, Germany
M. Jolis. Universitat Autonoma de Barcelona. Spain
R. Leandre. Universite Louis Pasteur. Strasbourg, France
J.F. Le Gall. Universite de Paris VI. France
P. Malliavin. Universite de Paris VI. France
E. Mayer-Wolf. Technion. Haifa, Israel
A. Millet. Universite de Paris X. France
D. Nualart. Universitat de Barcelona. Spain
D. Ocone. Rutgers University. U.S.A.
E. Pardoux. Universite de Provence. France
G. Da Prato. Scuola Normale Superiore. Pisa, Italy
B. 0ksendal. University ofOslo. Norway
J. San Martin. Universidad Cat6lica de Chile. Santiago, Chile
M. Sanz-SoIe. Universitat de Barcelona. Spain
A. Sintes. Universitat Autonoma de Barcelona. Spain
J. L. Sole. Universitat Politecnica de Catalunya. Spain
A. S. Ustiinel. E.N.S.T. Paris, France
F. Utzet. Universitat de Barcelona. Spain
J. Van Biesen. Universiteit Antwerpen. Wilrijk, Belgium
J. Vives. Universitat Autonoma de Barcelona. Spain
R.J. Williams. University ofCalifornia, San Diego. U.S.A.
M. Wschebor. Universidad de la Republica. Montevideo, Uruguay
J. Zabczyk. Polish Academy ofSciences. Poland
M. Zakai. Technion. Haifa, Israel
O. Zeitouni. Technion. Haifa, Israel
CONTENTS
Paolo Baldi and Marta Sanz-Sole
Modulus ofContinuity for Stochastic Flows 1
Rainer Buckdahn
Nonlinear Skorohod Stochastic Differential Equations 21
Ana Bela Cruzeiro and Jean-Claude Zambrini
Ornstein-Uhlenbeck Processes as Bernstein Processes 40
Luis G. Gorostiza and Alfredo L6pez-Mimbela
A Convergence Criterion for Measure-Valued Processes,
and Application to Continuous Superprocesses 62
Remi Leandre
A simple prooffor a large deviation theorem 72
Paul Malliavin
Universal Wiener Space 77
Annie Millet and Marta Sanz-Sole
On the Support ofa Skorohod Anticipating Stochastic
Differential Equation 103
David Nualart and Moshe Zakai
Positive and Strongly Positive Wiener Functionals 132
Daniel L. Ocone
A Symmetry Characterization ofConditionally Independent
Increment Martingales 147
Bernt 0ksendal and Tu-Sheng Zhang
The Stochastic Volterra Equation 168
L. A. Shepp and Ofer Zeitouni
Exponential Estimates for Convex Norms and Some
Applications 203
Ruth J. Williams
Reflected Brownian Motion: Hunt Processes
and Semimartingale Representation 216
Jerzy Zabczyk
The Fractional Calculus and Stochastic Evolution Equations 222
Modulus of Continuity for Stochastic Flows
Paolo Baldi and Marta Sanz-Sole
(*) (**)
Abstract. In this article we determine the modulus ofcontinuityfor aclass
of stochastic flows. We also give an application to anticipating stochastic
differential equations ofthe Stratonovich type.
o.
INTRODUCTION
The purpose of this paper has been to prove a result on the modulus of
continuity ofthe stochastic flow on IRd
tE[O,I], (0.1)
=
withrespect to the Riemannian metricdassociated witha(x) a(x)a*(x),
in the strictly elliptic case.
More explicitely, for any compact set A of IRd we define the modulus
ofcontinuity of<p as
and we prove (see Theorem 2.1) that
An analogous result for fixed initial condition x has been proved in [2,3].
Here we use the method introduced in [3], that means our main result will
be obtained as a consequence of some large deviation estimates, uniform
with respect to x.
Themotivationofourworkhasbeentostudypathpropertiesoftheso
lutionofananticipatingstochasticdifferentialequationofthe Stratonovich
type like that studied in [11]. The solution ofthis class ofequations is ob
tained by the composition ofthe adapted flow (0.1) with the non adapted
(*) Thisworkwas partiallydone while the authorwas visitingthe "Centrede Recerca
Matematica".
(**) Partially supported by a grant ofthe DGICYT nO PB 90-0452.
2 P. Baldi and M. Sanz-SoIe
initial condition, and for this reason the path properties ofthe flow can be
transferred to their analogues for the solution ofthe anticipating equation.
The structure of the paper is as follows. Section 1 gives the background
on large deviations estimates which is needed in the sequel. The most im
portant result gives an extension of Azencott's continuity lemma for the
flow, using tubes depending on E. The proof uses an integration by parts
technique developped in [7] and someexponentialinequality proved in [10],
(see also [12]). A particular case of this result has been established in [9]
using Sovolev's inequalities. Section 2 is devoted to prove the announced
result on the modulus of continuity for the flow. Finally in Section 3, we
present an application to the study of the modulus of continuity of the
Ocone-Pardoux's anticipating Stratonovichstochastic differentialequation.
1. AN EXTENSION OF AZENCOTT'S CONTINUITY
LEMMA FOR THE FLOW
Let h: R+ --. R+ be a function which is bounded on compact subsets of
R+ and such that
lim h(E) = 0 .
e->O E
We consider a family ere , er :Rd --. IRd18'Ilk , be , b:Rd --. Rd, E>0 of
measurable functions satisfying the following hypotheses:
(HI) (a) ere are ofclass C2 and be are ofclass C1, E >O.
(b) er and b are locally Lipschitz, and there exists a constant M >0
such that Ier(x)I+Ib(x)I~ M( 1+Ixl), for any x E Rd.
(c) lim h(l) {W(y) - b(y)I+Iere(y) - cr(y) I} = 0, uniformly on
e->O E
compact subsets of Rd.
Denote by 1{k the set of functions I : [0,1] --. Rk with 1(0) = 0 which
are absolutely continuous and such that Jo1IIs. 12ds < +00. For I E 1{k
we set II I Ilk = ( JolI'Is 12ds) 1/2.
In this section we will consider the stochastic differential equations on Rd
it it
iPHx) =x+E ere (iP~(x)) dWs + be (iP~(x)) ds , (1.1)
x E IRd, E >0, defined on the canonical Wiener space associated with the
k-dimerisional Wiener process W.
Let I E 1{k, and consider the ordinary differential equation