A duality approach for the weak approximation of s

arXiv:math/0610178v1[math.PR]5Oct2006TheAnnalsofAppliedProbability2006,Vol.16,No.3,1124–1154DOI:10.1214/105051606000000060cInstituteofMathematicalStatistics,2006ADUALITYAPPROACHFORTHEWEAKAPPROXIMATIONOFSTOCHASTICDIFFERENTIALEQUATIONSByEmmanuelleCl´ement,ArturoKohatsu-Higa1andDamienLambertonUniversit´edeMarne-la-Vall´ee,OsakaUniversityandUniversit´edeMarne-la-Vall´eeInthisarticlewedevelopanewmethodologytoproveweakap-proximationresultsforgeneralstochasticdifferentialequations.In-steadofusingapartialdifferentialequationapproachasisusuallydonefordiffusions,theapproachconsideredhereusesthepropertiesofthelinearequationsatisfiedbytheerrorprocess.Thismethodol-ogyseemstoapplytoalargeclassofprocessesandwepresentasanexampletheweakapproximationofstochasticdelayequations.1.Introduction.TheEulerschemeforstochasticdifferentialequationsiswidelyusedinapplicationsasitiseasytocompute.TheEulerschemecanbeeasilygeneralizedtoavarietyofstochasticequationsbeyondtheframeworkofdiffusionequations,inparticularVolterraSDEs,delaySDEs,anticipatingSDEsandnonlinearSDEs.Ontheotherhand,thetheoreticalpropertiesoftheEulerschemearemostlystudiedforthediffusioncaseasmostoftheresultsavailablesofarareinthisframework.Insomecases,extensionstoothersimilarequationsarestraightforwardbutinothercases,additionalnontrivialworkisrequired.Forexample,see[8]forextensionstosemimartingales,and[1,11]forap-proximationsofanirregularfunctionalofadiffusionwhichisapproachedusingaEulertypescheme.ItisalsowellknownthatthedefinitionofanextensionoftheEulerschemefordelaytypesystemsisstraightforwardbutthetechnicalresultsontheweakrateofconvergencearelimited.See[4,6,9].Inthisarticleweproposeageneralizationofthetheoryofweakapproxi-mationswhichstudiestherateofconvergenceoftheEulerschemeconsideredReceivedDecember2004;revisedOctober2005.1SupportedinpartbyGrantsBFM2003-03324andBFM2003-04294.AMS2000subjectclassifications.60H07,60H10,60H35,65C30.Keywordsandphrases.Stochasticdifferentialequation,weakapproximation,Eulerscheme,Malliavincalculus.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofAppliedProbability,2006,Vol.16,No.3,1124–1154.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12E.CL´EMENT,A.KOHATSU-HIGAANDD.LAMBERTONinlaw.Thisgeneralizationfindsasanapplicationtheweakrateofconver-genceofsmoothfunctionalsofgeneraldelaytypesystemsandalsocovers,withafurtherstudyoftheMalliavincovariancematrix,thecaseofirregularfunctionsofthesolutionofthestochasticequation.Themainideaistochangecompletelytheapproachuseduntilnowtoproveweakapproximationrateresults.Thisnewidea,whichusesthewholepathoftheprocessunderstudyratherthanthepartialdifferentialequationassociatedtotheproblem,shouldallowtoobtainvariousotherstraightfor-wardgeneralizationsofresultsoftheweakrateofconvergence.Inordertodescribeourapproachroughly,letXdenotethesolutionofastochasticequationand¯XtheEulerschemeassociatedtoit.TheproblemofweakrateofconvergenceconsistsinfindingtherateatwhichE(f(X)−f(¯X))convergestozeroforvariousclassesoffunctionalsf.Theoptimalrateisthestepsizeoftheschemeeventhoughtheequationsconsideredmaydiffer.Theclassicalproofofthisresultfordiffusionsisbasedontheassociatedpartialdifferentialequation,thatis,Ef(X)hasthroughtheFeynman–KacformulaaninterpretationusingPDEs.Thisistheimportantpointintheclassicalapproachwhichisnotusedinourapproach.Inthecaseofsomestochasticequations,iffisregularenough,theproofissimilariftheas-sociatedPDEexists.Iffisanirregularfunction,thentheissueofthenondegeneracyoftheMalliavincovariancematrixoftheEulerschemebe-comesanimportantissueashasbeenshownin[1,11],butthisextensionisnontrivial.Inthisarticleweproposeacompletelydifferentmethodtoproveweakapproximationresultsbasedonapathwiseapproach.Thatis,weusethemeanvaluetheoremtorewritef(X)−f(¯X)=R10f′(aX+(1−a)¯X)da(X−¯X).Then,wederivealinearequationsatisfiedbyY=X−¯X.Whenthisequationcanbeexplicitlysolved,whichseemstobetrueonlyfordiffusions,onecanobtaintherateofconvergencebyusingthedualitypropertyofstochasticintegrals.ThismethodologywasfirstintroducedbyKohatsu-HigaandPettersson[7]andusedinGobetandMunos[5].ItseemstobequitegeneralexceptfortheexplicitexpressionforYwhichcanbedoneonlyinthecaseofdiffusions.Thisarticlepresentsageneralframeworktoanalyzeweakapproximationsinstochasticequations.Inparticularwesolvetheproblemwithouthavinganexplicitexpressionforthesolutionofstochasticlinearequations,byusingadualityargument.Thisdualityformula(seeSection3)showsexplicitlytheweakerrorasaby-productoftheexpectationofanerrorprocess(calledGinSection3).Tofinishtheproofonehastousethedualityformulaforstochasticintegrals.Thereforeourapproachworksmostlyforstochasticequationswithregularcoefficients.Forthisreasonwehavetostudythestochasticderivativesofthesolutionprocess.ItshouldbeWEAKAPPROXIMATIONOFSDES3emphasizedthatthisapproachappliestoregularfunctionalsoftheprocessXandnotonlytofunctionsofthevalueofXatafixedtimet.Furthermore,theframeworkintroducedherealsoextendsnaturallytothecaseofirregularfunctionsf.Thatis,oneusestheintegrationbypartsformulaofMall