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...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 .展开更多
基金supported by National Natural Science Foundation of China(Grant No.11171062)Innovation Program of Shanghai Municipal Education Commission(Grant No.12ZZ063)
文摘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 .
