m-order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index

Given an integer m, a probability measure ν on [0, 1] a process X and a real function g, we define the m-order ν-integral having as integrator X and as integrand g. In the case of the fractional Brownian motion B^H, for any locally bounded function g, the corresponding integral vanishes for all odd indices m > 1/2H and any symmetric ν. One consequence is an Itô-Stratonovich type expansion for the fractional Brownian motion with arbitrary Hurst index H ∈]0, 1[. On the other hand we show that the classical Itô-Stratonovich formula holds if and only if H > 1/6.