中文摘要：

假设B是一个指数为H∈（0,1）,K∈（,1]且满足2HK〈1的双分数Brownian运动,其赋权局部时设为{L（x,t）,t≥0,x∈R}。建立了f（B）与B的广义二次协变差[f（B）,B]（W）,并且研究如下局部时的积分∫R f（x）L（dx,t）,t≥0,这里x｜→f（x）为Borel可测函数。构造了一个Banach空间H使得广义二次协变差在L2中存在,并且如下广义Bouleau-Yor型等式成立：[f（B）,B]t（W）=-21-K∫R f（x）L（dx,t）,t≥0,任意f∈H.藉此建立了一类其导数属于H时的绝对连续函数的广义Ito公式,作为应用给出了一类双分数Brownian运动的Ito-Tanaka公式。

英文摘要：

Let B be a bi-fractional Brownian motion with indices H∈（0,1）,K∈（0,1] such that 2HK1,and let {L（x,t）,t≥0,x∈R} be its local time process.The generalized quadratic covariation [f（B）,B]（W） of f（B） and B is introduced.The integral ∫Rf（x）L（dx,t）, t≥0is studied,where x｜→f（x） is a Borel measurable function.A Banach space H is constructed,which satisfies the generalized quadratic covariation exists in L2 and the generalized Bouleau-Yor identity takes the form [f（B）,B]（W）t=-21-K∫Rf（x）L（dx,t）, t≥0for all f∈H.Thereby,the generalized Ito formula for absolutely continuous functions with derivatives belonging to H is investiagted.As an application,the Ito-Tanaka formula for the bi-fractional Brownian motion with 2HK1 is obtained.