The generalized Bouleau-Yor identity for a sub-fractional Brownian motion  

作　　者：YAN LiTan HE Kun CHEN Chao 

机构地区：[1]Department of Mathematics, College of Science, Donghua University [2]Department of Mathematics, East China University of Science and Technology

出　　处：《Science China Mathematics》2013年第10期2089-2116,共28页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11171062);Innovation Program of Shanghai Municipal Education Commission(Grant No.12ZZ063)

摘　　要：Let SH be a sub-fractional Brownian motion with index 0 〈 H 〈 1/2. In this paper we study the existence of the generalized quadratic eovariation [f（SH）, SH]（W） defined by[f（SH）,SH]t（W）=lim ε↓0 1/ ε2H ∫t 0 {f（SH s＋ε）-f（SH s＋ε）-f（SH s）}（SH s＋ε -SH s）ds2H, provided the limit exists in probability, where x → f（x） is a measurable function. We construct a Banach space X of measurable functions such that the generalized quadratic covariation exists in L2 provided f ∈ X. Moreover, the generalized Bouleau-Yor identity takes the form -∫R f（x） H（dx,t）=（2-2 2H-1）[f（SH ）,SH]t（w） for all f ∈ where H （X, t） is the weighted local time of SH. This allows us to write the generalized ItS＇s formula for absolutely continuous functions with derivative belonging to .Let SH be a sub-fractional Brownian motion with index 0 < H < 1/2.In this paper we study the existence of the generalized quadratic covariation [f(SH),SH](W) defned by[f(SH), SH](W)t= limε→01/ε2H∫t0{f(SHs+ε)-f(SHs)}(SHs+ε-SHs)ds2H,provided the limit exists in probability, where x → f(x) is a measurable function. We construct a Banach space ■ of measurable functions such that the generalized quadratic covariation exists in L2 provided f ∈ H.Moreover, the generalized Bouleau-Yor identity takes the form ∫Rf(x)ψH(dx, t) =(2-22H-1)[f(SH), SH](W)t for all f∈■, where ψH(x, t) is the weighted local time of SH. This allows us to write the generalized It's formula for absolutely continuous functions with derivative belonging to

关 键 词：sub-fractional Brownian motion Malliavin calculus local time Ito＇s formula quadratic covaria-tion 

分 类 号：O211.6[理学—概率论与数理统计] O177.2[理学—数学]