Stochastic derivatives for fractional diffusions

arXiv:math/0604315v4[math.PR]18Oct2007TheAnnalsofProbability2007,Vol.35,No.5,1998–2020DOI:10.1214/009117906000001169cInstituteofMathematicalStatistics,2007STOCHASTICDERIVATIVESFORFRACTIONALDIFFUSIONSByS´ebastienDarsesandIvanNourdinUniversit´edeFranche-Comt´eandUniversit´eParis6Inthispaper,weintroducesomefundamentalnotionsrelatedtotheso-calledstochasticderivativeswithrespecttoagivenσ-fieldQ.Inourframework,werecallwell-knownresultsaboutMarkov–Wienerdiffusions.WethenfocusmainlyonthecasewhereXisafractionaldiffusionandwhereQisthepast,thefutureorthepresentofX.WetreatsomecrucialexamplesandourmainresultistheexistenceofstochasticderivativeswithrespecttothepresentofXwhenXsolvesastochasticdifferentialequationdrivenbyafractionalBrow-nianmotionwithHurstindexH1/2.Wegiveexplicitformulas.1.Introduction.Thereexistvariouswaystogeneralizethenotionofdif-ferentiationondeterministicfunctions.Wemaythinkoffractionalderiva-tivesordifferentiationinthesenseofthetheoryofdistributions.Inbothcases,weloseadynamicalorgeometricinterpretationoftangentvectors(velocities,e.g.).Inthepresentwork,weseektoconstructderivativesonstochasticprocesseswhichconserveadynamicalmeaning.Ourgoalismoti-vatedbythestochasticembeddingofdynamicalsystemsintroducedin[2].Thisprocedureaimsatcomprehendingthefollowingquestion:howcanwewriteanequationwhichcontainsthedynamicalmeaningofaninitialor-dinarydifferentialequationandwhichextendsthisdynamicalmeaningtostochasticprocesses?Wereferto[3]formoredetails.Unfortunately,formostofthestochasticprocessesusedinphysicalmod-els,thelimitZt+h−Zthdoesnotexistpathwise.Whatcanbedonetogiveameaningtothislimit?Oneofthemaintoolsavailableisthe“quantityofinformation”whichwecanusetocalculateit,namelyagivenσ-fieldQ.TheideaisthatonecanReceivedApril2006;revisedSeptember2006.AMS2000subjectclassifications.Primary60G07,60G15;secondary60G17,60H07.Keywordsandphrases.Stochasticderivatives,Nelson’sderivative,fractionalBrownianmotion,fractionaldifferentialequation,Malliavincalculus.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofProbability,2007,Vol.35,No.5,1998–2020.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12S.DARSESANDI.NOURDINremovethedivergenceswhichappearpathwisebyaveragingoverabundleoftrajectoriesinthepreviouscomputation.ThisfactcanbeachievedbystudyingthebehaviorwhenhgoestozerooftheconditionalexpectationEZt+h−ZthQ.SuchobjectswereintroducedbyNelsoninhisdynamicaltheoryofBrowniandiffusion[9].Forafixedtimet,hecalculatesaforward(resp.,backward)derivativewithrespecttoagivenσ-fieldPtwhichcanbeseenasthepastoftheprocessuptotimet(resp.,Ft,thefutureoftheprocessaftertimet).ThemainclasswithwhichthiscanbedoneturnsouttobethatofWienerdiffusions.Thepurposeofthispaperis,ononehand,tointroducenotionswhichcanbeusedtostudytheaforementionedquantitiesforgeneralprocessesand,ontheotherhand,totreatsomeexamples.WemainlystudythesenotionsonsolutionsofstochasticdifferentialequationsdrivenbyafractionalBrownianmotionwithHurstindexH≥12.Inparticular,werecallresultsonWienerdiffusions(caseH=12)inourframework.Weprovethatforasuitableσ-algebra,thestochasticderivativesofasolutionofthefractionalstochasticdifferentialequationexistandweareabletogiveexplicitformulas.Ourpaperisorganizedasfollows.InSection2,werecallsomenowclas-sicalfactsonstochasticanalysisforfractionalBrownianmotion.InSection3,weintroducethefundamentalnotionsrelatedtotheso-calledstochasticderivatives.InSection4,westudystochasticderivativesofNelson’stypeforfractionaldiffusions.WeshowinSection5thatstochasticderivativeswithrespecttothepresentturnouttobeadequatetoolsfortreatingfractionalBrownianmotionwithH12.Wealsotreatthemoredifficultcaseofafractionaldiffusion.2.BasicnotionsforfractionalBrownianmotion.WebrieflyrecallsomebasicfactsconcerningstochasticcalculuswithrespecttoafractionalBrown-ianmotion;referto[12]forfurtherdetails.LetB=(Bt)t∈[0,T]beafractionalBrownianmotionwithHurstparameterH∈(0,1)definedonaprobabilityspace(Ω,F,P).ThismeansthatBisacenteredGaussianprocesswiththecovariancefunctionE(BsBt)=RH(s,t),whereRH(s,t)=12(t2H+s2H−|t−s|2H).(1)IfH=1/2,thenBisaBrownianmotion.From(1),onecaneasilyseethatE|Bt−Bs|2=|t−s|2H,soBhasα-H¨oldercontinuouspathsforanyα∈(0,H).STOCHASTICDERIVATIVESFORFRACTIONALDIFFUSIONS32.1.Spaceofdeterministicintegrands.WedenotebyEthesetofstepR-valuedfunctionson[0,T].LetHbetheHilbertspacedefinedastheclosureofEwithrespecttothescalarproducth1[0,t],1[0,s]iH=RH(t,s).Wedenoteby|·|Htheassociatednorm.Themapping1[0,t]7→Btcanbeex-tendedtoanisometrybetweenHandtheGaussianspaceH1(B)associatedwithB.Wedenotethisisometrybyϕ7→B(ϕ).WhenH∈(12,1),itfollowsfrom[14]thattheelementsofHmaynotbefunctionsbutdistributionsofnegativeorder.ItwillbemoreconvenienttoworkwithasubspaceofHwhichcontainsonlyfunctions.Suchaspaceistheset|H|ofallmeasurablefunctionsfon[0,T]suchthat|f|2|H|:=H(2H−1)ZT0ZT0|f(u)||f(v)||u−v|2H−2dudv∞.Weknowthat(|H|,|·||H|)isaBanachspace,butthat(|H|,h·,·iH)isnotcomplete(see,e.g.,[14]).Moreover,wehavetheinclusionsL2([0,T])⊂L1/H([0,T])⊂|H|⊂H.(2)2.2.Fractionaloper