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

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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 of the material is concemed, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to <,verwertungsgesellschaft Wort», Munich. © 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

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