m-order integrals and generalized Ito'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(X). In the case of the fractional Brownian motion B, 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 01/6.