STOCHASTIC DIFFERENTIAL EQUATIONS WITH FRACTAL NOI

STOCHASTICDIFFERENTIALEQUATIONSWITHFRACTALNOISEM.ZähleMathematicalInstitute,UniversityofJena,D-07740Jena,GermanyE-mail:[e-mail]fferentialequationsinRnwithrandomcoefficientsareconsid-eredwhereonecontinuousdrivingprocessadmitsageneralizedquadraticvariationprocess.ThelatterandtheotherdrivingprocessesareassumedtopossesssamplepathsinthefractionalSobolevspaceWβ2forsomeβ1/2.Thestochasticintegralsaredeterminedasanticipatingforwardintegrals.ApathwisesolutionprocedureisdevelopedwhichcombinesthestochasticItôcalculuswithfractionalcalculusvianormestimatesofassociatedinte-graloperatorsinWα2for0α1.Linearequationsareconsideredasaspecialcase.ThisapproachleadstofastcomputeralgorithmsbasingonPicard’sitera-tionmethod.Keywords:forwardintegral,quadraticvariationprocess,anticipatingstochasticdifferentialequation,BesovspaceMRClassification:60H10,60H05,60H15,34F051IntroductionForapplicationsinmathematicalfinance(see,e.g.[10])andotherfieldsthefollowingstochasticdifferentialequationinRnisrelevant:dX(t)=a0¡X(t),t¢dW(t)+mXj=1aj¡X(t),t¢dBHj(t)+b¡X(t),t¢dtX(t0)=X0.HereWistheone–dimensionalWienerprocessandtheBHjdenotefractionalBrownianmotionswithHurstexponentsHjwhichmayvaryintime.TheymaybeintroducedbymeansoftherepresentationBHj(t):=Reitu−1|u|Hj(t)+1/2dWj(u)forWienerprocessesW1,...,Wm.(ForconstantHjweobtainclassicalfractionalBrownianmotion.)WewillconsideronlythecaseHj(t)≥H12andassumethatallHj(t)areHöldercontinuousfunctions.InthiscasethesamplepathsoftheBHjhavealmostsurelynicefractionalsmoothnessproperties:TheyareHöldercontinuousofallorderslessthanHandtheypossessfinitepvariations,p1H,andfractionalderivativesofallorderslessthanH.Moreover,theyareelementsoftheSobolev-Slobodeckij(orBesov)spacesWH−2whicharemostap-propriatetoourapproach.Thisguaranteesthatundersmoothnessassumptionsontherandomvectorfieldsaithestochasticintegralsinthesecondsummandoftheaboveequationforasuitablenotionofsolutionmaybeunderstoodinthesenseofa.s.convergenceoftheRiemann–Stieltjessums.Thestochasticintegralforthefirstsummand,i.e.theBrownianmotioncomponent,maybedeterminedinthesenseofuniformconvergenceinprobabilityoftheRiemann–Stieltjessums.(Wedonotassumeanyadaptednessontherandomvectorfieldsa0,a1,...,am,b.)Inparticular,thecorrespondinglinearequationmaybeusedformodellingstockpricedevelopmentsandoptionpricinginmathematicalfinance.ThefractionalBrownianmotioncomponentsleadtolongrangedependenceandtheWienerprocessguaranteesstrongno-arbitrage(see[15]).Ingeneral,theaboveequationcannotbetreatedbymeansofsemimartingaletheory(foradaptedcoefficients).BecauseoftheBrownianmotioncomponentitalsodoesnotfitintothemodelsusingfinitep-variations,Hölderconditions,orfractionalderivatives,wherethesolutiondependscontinuouslyonthedrivingprocesses.Thesesupposeintegrandsandintegratorsofsummedorderoffrac-tionalsmoothnessgreaterthan1.InrecentmoreabstractpapersthecaseoflowerorderHurstexponentshasbeenconsideredfordifferenttypesofstochas-ticintegrals,butthesolutionproceduresarecomplicated.Anumericallymoreaccessibleapproachtothepathwisesolutionconsistsinthefollowing.(Forsim-plicityitisdemonstratedhereonthetimehomogeneouscase,thegeneralcasewillbetreatedbelow.)AsintheDoss-Sussmanapproachform=0weseekthesolutionpathwiseintheformX(t)=h(Y(t),W(t))2forsomesmoothfunctionhsatisfying∂h∂y(y,z)=a0(h(y,z))andanunknownvectorprocessY(t).ThenweapplytheItôformulawhichwillbeadoptedasageneralcalculationruleinordertodetermineanauxiliarystochasticdifferentialequationwheredW(t)iseliminated:dY(t)=mXi=1˜ai(Y(t),W(t))dBHi(t)+˜b(Y(t),W(t))dtwithnewrandomvectorfields˜a1,...˜am,˜b.HereW(t)maybeconsideredasaparameterfunction(oflessorderofsmoothnessthanthatoftheintegrators).Suchtypeofequationshavebeentreatedinourpaper[13]inamorespecialsituation.Extendingthisapproachtomappingsdependingonparameterfunc-tionsasabovewegetpathwiseauniquelocalsolutionY(t)withcoordinatesintheBesovspaceWH−2.SubstitutingthisY(dependingonthechoiceofh)intheaboveformulaforX(t)weindeedobtainasolution.WewillshowthatthesolutionisuniqueintheclassofallprocesseswithgeneralizedquadraticvariationssatisfyingtheItôcalculationrule.(Fordifferenthweobtaindifferentrepresentationsofthesameprocess.)Notethatthe(random)functionhandtheauxiliaryprocessYmaybedeterminedbyPicard’siterationmethod.TheessentialpartofourapproachistheaboveauxiliarySDEfortheprocessY.ItissolvableformoregeneraldrivingprocessesandparameterprocessesthanBH1,...,BHmandW,resp.Moreover,themethodofshowingconvergenceinprobabilityoftheRiemann–StieltjessumsforthefirstintegralintheaboveequationbymeansoftheTaylorexpansioniswell–knownfromtheliteratureandgoesbacktoFöllmer[5].Incontrasttootherpaperswederivethisconvergencefromthatoftheremainingintegrals.Thelattercanbeshownbyatleastthreedifferentmethodsusingpathwisefractionalsmoothnesspropertiesofintegrandsandintegratorsofsummedordergreaterthan1(cf.Section2.1).TheWienerprocessWwillbereplacedbyanarbitrarycontinuousprocessZwithgeneralizedquadraticvariation[Z]∈WH−2ifthefirstintegralisunderstoodinasimilarsense.InsteadoffractionalBrownianmotionsBHwewillchoosearbitraryprocess