Generalized covariations, local time and Stratonovich Ito's formula for fractional Brownian motion with Hurst index H ≥ 1/4

Given a locally bounded real function g, we examine the existence of a 4-covariation [g(B-H), BH, BH, BH], where BH is a fractional Brownian motion with a Hurst index H greater than or equal to (1)/(4). We provide two essential applications. First, we relate the 4-covariation to one expression involving the derivative of local time, in the case H = (1)/(4), generalizing an identity of Bouleau-Yor type, well known for the classical Brownian motion. A second application is an Ito formula of Stratonovich type for f (B-H). The main difficulty comes from the fact BH has only a finite 4-variation.