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Stochastic Analysis and Related Topics: In Honour of Ali Süleyman Üstünel, Paris, June 2010 PDF

222 Pages·2012·3.17 MB·English
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Springer Proceedings in Mathematics and Statistics Volume 22 Forfurthervolumes: http://www.springer.com/series/10533 Springer Proceedings in Mathematics and Statistics Thisbookseriesfeaturesvolumescomposedofselectcontributionsfromworkshops and conferences in all areas of current research in mathematics and statistics, including OR and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher,individualcontributionsare allrefereedto the high qualitystandardsof leadingjournalsinthefield.Thus,thisseriesprovidestheresearchcommunitywith well-edited, authoritative reports on developments in the most exciting areas of mathematicalandstatisticalresearchtoday. Laurent Decreusefond (cid:2) Jamal Najim Editors Stochastic Analysis and Related Topics ¨ In Honour of Ali Su¨leyman Ustu¨nel Paris, June 2010 123 Editors LaurentDecreusefond JamalNajim TelecomParisTech Paris,France ISSN2194-1009 ISSN2194-1017(electronic) ISBN978-3-642-29981-0 ISBN978-3-642-29982-7(eBook) DOI10.1007/978-3-642-29982-7 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012943599 MathematicalSubjectClassification(2010):60H07,60H15,60G22,60G55 (cid:2)c Springer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Foreword The present contributed volume resulted from the “9th Workshop on Stochastic Analysis and Related Topics,” part of a series of biannual workshopsinitiated by H. Ko¨rezlioglu and A.S. U¨stu¨nel in 1986 and then continued with the help of B.Øksendaluntil2003.Thisevent,heldonthe14thand15thofJune,2010,wasan idealoccasiontocelebratethe60thbirthdayofA.S.U¨stu¨nelandhiscontributions tomathematics. We would like to thank the Institut Telecom and Ge´rard Memmi, head of the “ComputerScienceandNetworking”department,whofullysponsoredthisevent. Paris,France L.Decreusefond J.Najim v • Preface After brilliant studies in the most renowned turkish institutions, Ali Suleyman U¨stu¨nelwaslongingtobecomeaphysicistwhenHayriKo¨rezliogluconvincedhim toswitchtomathematics.Thiswasthebeginningofalonganddeepcollaboration andfriendship. A.S. U¨stu¨nel finally defended his Ph.D. in probability in Paris in 1981 with LaurentSchwarzasanexaminer.HefirstbegantoworkatCentreNationald’E´tudes enTe´le´communications(nowOrangeLabs)andthenatEcoleNationaleSupe´rieure des Te´le´communications (now Te´le´com Paristech). His first works were related to nuclear-valued processes. The strong topological properties of nuclear spaces inducethatmanypropertiesonlyhavetobeverified“cylindrically”to holdinfull generality:Forinstance,aprocess.X.t/; t (cid:3)0/withvaluesinthesetoftempered distributions is continuous if and only if for any ' rapidly decreasing, the real- valued process .<X.t/; ' >; t (cid:3) 0/ is continuous. The work of A.S. U¨stu¨nel culminated in the “three operators lemma” which states that when three Hilbert– Schmidtoperatorsareappliedinarowtoacylindricalsemi-martingale,itbecomes atruesemi-martingale. In the mid-1980s, he was one of the pioneering researchers to investigate thoroughlythenewlybornMalliavinCalculus,afieldwherehequicklybecame(and stillis!)aworldrenownedexpert.From1986,H.KorezliogluandA.S.U¨stu¨nelorga- nizedthe“Stochasticanalysisandrelatedtopics”worskhopwhosefirstoccurrences tookplaceinSilivri(Turkey)every2years.The“Silivriband”(mainlyM.Chaleyat- Maurel,A.Grorud,A.Millet,D.Nualart,E.Pardoux,M.Pontier,M.Sanz)played a major role in the developmentof Malliavin calculus and its applications. At the same time, A.S. U¨stu¨nel and Moshe Zakai started a collaboration which was to lastforthenext20years.Theirmainsubjectofinvestigationhasbeentheabsolute continuityofshifttransformationsintheWienerspace.Itiswellknownthatthelaw ofBrownianmotionwithanadapted,squareintegrabledriftisabsolutelycontinuous with respectto the law of the Brownianmotion.Theydevotedtheir whole energy toextendthefamilyofadmissibledrifts,thatistosaydriftssuchthattheabsolute continuity propertystill holds. The main question is to get rid of the adaptability. They showed that this can be replaced by, for instance, either monotony or some vii viii Preface regularityontheMalliavinderivativeofthedrift.Mostoftheirresultsarecontained inthebeautifulbooktheycoauthored. Thereciprocalproblemcanbeinformallystatedas:Givenameasureequivalent to the Wiener measure, does there exist a shift transformation which realizes this measure?It turnedoutthatthe optimaltransportationtheorywhichwas regaining interest after the work of Brenier in the end of the previous millenium yielded an answer to this problem. Using his previously defined notion of H-convexity, A.S. U¨stu¨nel, in collaboration with Denis Feyel, solved the so-called Monge– Kantorovitch problem in the Wiener space for the Cameron–Martin cost. Once again, Malliavin calculus provided the convenient concepts to generalize almost word for word, the results known in finite dimension. Surprisingly, the proofs of someresultssuchasTalagrandorPoincare´inequalitiesappearedtobeevensimpler ininfinitedimensionduetotheavailabilityoftheItoˆcalculus.Inseveralpapers,they showed different properties of the solution of the Monge–Kantorovitch problem, whichyieldedinturnseveralfunctionalinequalities. Combining all his earlier results, A.S. U¨stu¨nel founda criterion which ensures theinvertibilityofashifttransformationontheWienerspace:Ifthekineticenergy of the driftu is equalto the entropyof the measureinducedby the corresponding shifttransformation,then the map! 7! ! Cu.!/ is invertible.Such a resultcan be interpreted as a construction of a strong solution of the stochastic differential equationdX.t/D(cid:4)uP.X;t/dtCdB.t/forverygeneralu. ThisquickglanceatA.S.U¨stu¨nel’sworkdoesnotgivejusticetohisothernumer- ous contributions to control, filtering, functional inequalities, fractional Brownian motion, etc. but it shows a strong line of thought and a constant will to focus on deep problems. To borrow one of his favorite metaphor:Instead of looking to the holewhich correspondsto the key,he ratherprefersto seek forthe key whichfits intothehole. Besides his own research activities, A.S. U¨stu¨nel has been the professor of severalgenerationsofstudentsatTelecomParisTechandthePh.D.advisorofmany students. We have all been impressed by notonly his passion to mathematics, his wide knowledge, but also his generosity, his kindness and the relevance of his advice. On a more artistic side, Su¨leyman and his wife, Jacqueline, became world- renowned specialists of the Turkish painter Fikret Moualla, many paintings of whom can be seen at their galleryin Paris. But that is anotherstory. We take this opportunity to deeply thank Jacqueline, whose help was crucial to organize this workshop,ormoreprecisely,toconvinceSu¨leymantoparticipateinthisworkshop organizedontheoccasionofhis60thbirthday. HappybirthdaySu¨leyman! Paris,France L.Decreusefond J.Najim Contents 1 ALook-DownModelwithSelection ....................................... 1 BoubacarBah,EtiennePardoux,andAhmadouBambaSow 1.1 IntroductionandPreliminaries ......................................... 1 1.2 Wright–FisherDiffusionwithSelectionandDuality................. 3 1.3 Look-DownwithSelection,Exchangeability.......................... 5 1.4 ConvergencetotheWright–FisherDiffusionwithSelection......... 14 References..................................................................... 28 2 ControlofInventorieswithMarkovDemand ............................ 29 AlainBensoussan 2.1 Introduction ............................................................. 29 2.2 NoBacklogandNoSet-UpCost....................................... 30 2.3 NoBacklogandSet-UpCost........................................... 47 References..................................................................... 55 3 On the Splitting Method for Some Complex-Valued QuasilinearEvolutionEquations........................................... 57 ZdzisławBrzez´niakandAnnieMillet 3.1 Introduction ............................................................. 57 3.2 Well-PosednessandFirstAPrioriEstimates.......................... 60 3.3 SpeedofConvergence.................................................. 73 3.4 SpeedofConvergencefortheSplittingMethod....................... 86 References..................................................................... 89 4 AModelizationofPublic–PrivatePartnershipswithFailureTime.... 91 CarolineHillairetandMoniquePontier 4.1 Introduction ............................................................. 92 4.2 TheProblemSetting .................................................... 92 4.3 SolutionoftheProblemWithoutDefault.............................. 94 4.4 IntroductionofaDefaultTime,WithoutPenalty...................... 101 4.5 PenaltyinCaseofDefaultwithr D0................................. 110 ix

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Since the early eighties, Ali Süleyman Üstünel has been one of the main contributors to the field of Malliavin calculus. In a workshop held in Paris, June 2010 several prominent researchers gave exciting talks in honor of his 60th birthday. The present volume includes scientific contributions f
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