The Generalized Quadratic Covariation For A Bi-fractional Brownian Motion And Related Topics

Let B be a bi-fractional Brownian motion with indices H ?(0,1),K E(0,1]such that 2HK<1,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,and the integral?R f(x)L(dx,t),t?0is studied,where x ? f(x)is a Borel measurable function.We construct a Banach space H,such that the generalized quadratic covariation exist in L2 for f ? H,and the generalized Bouleau-Yor identity takes the form (?),t? 0 hold for all f ? I.Thereby,the generalized Ito formula for absolutely continuous function with derivatives belonging to H is investigated.As an application,the Ito-Tanaka formula for the bi-fractional Brownian motion with 2HK<1 is obtained.