the generalized quadratic covariation for fractional brownian motion with hurst index less than 0.5

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1、THEGENERALIZEDQUADRATICCOVARIATIONFORFRACTIONALBROWNIANMOTIONWITHHURSTINDEXLESSTHAN1/2∗†,§‡♮LITANYAN,CHAOCHENANDJUNFENGLIU†DepartmentofMathematics,DonghuaUniversity2999NorthRenminRd.,Songjiang,Shanghai201620,P.R.China‡DepartmentofMathematics,EastChinaUniversityofScienceandTechnology130MeiLongRd.,Xu

2、hui,Shanghai200237,P.R.China♮DepartmentofMathematics,NanjingAuditUniversity,86WestYushangRd.,Nanjing211815,P.R.ChinaHAbstract.LetBbeafractionalBrownianmotionwithHurstindex0

3、(Bs+ε−Bs)sds,ε↓0ε0wherethelimitisuniforminprobabilityandx7→f(x)isadeterministicfunction.WeconstructaBanachspaceHofmeasurablefunctionssuchthatthegeneralizedquadratic2covariationexistsinLandtheBouleau-YoridentitytakestheformZHH(W)H[f(B),B]t=−f(x)L(dx,t)RHHprovidedf∈H,whereL(x,t)istheweightedlocaltime

4、ofB.ThisallowsustowritethefractionalItˆoformulaforabsolutelycontinuousfunctionswithderivativebelongingtoH.Thesearealsoextendedtothetime-dependentcase.1.IntroductionGivenH∈(0,1),afractionalBrownianmotion(fBm)withHurstindexHisameanarXiv:1106.2302v2[math.PR]18Jun2011zeroGaussianprocessBH={BtH,0≤t≤T}su

5、chthat1EBHBH=t2H+s2H−

6、t−s

7、2Hts2forallt,s>0.ForH=1/2,BHcoincideswiththestandardBrownianmotionB.BHisneitherasemimartingalenoraMarkovprocessunlessH=1/2,somanyofthepowerfultechniquesfromstochasticanalysisarenotavailablewhendealingwithBH.AsaGaussianprocess,onecanconstructthestochasticcalculusofvaria

8、tionswithrespecttoBH.SomesurveysandcompleteliteraturesforfBmcouldbefoundinBiaginietal[2],Decreusefond∗TheProject-sponsoredbyNSFC(10571025).§Correspondingauthor(litanyan@hotmail.com).2000MathematicsSubjectClassification.Primary60G15,60H05;Secondary60H07.Keywordsandphrases.fractionalBrownianmotion,Mal

9、liavincalculus,localtime,fractionalItˆofor-mula,quadraticcovariation.12L.YAN,C.CHENANDJ.LIUandUst¨unel[6],Gradinaru¨etal[14,15],Hu[18],Mishura[19]andNualart[23].Itiswell-knownthattheusualquadraticvariation[