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

222 Pages·2012·3.17 MB·English

by  Laurent Decreusefond

Springer Proceedings in Mathematics
and Statistics
Volume 22
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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
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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