Gaussian and non-Gaussian processes of zero power variation

We consider a class of stochastic processes X defined by X(t) = f₀^T G(t,s)dM(s) for t ∈ [0,T], where M is a square-integrable continuous martingale and G is a deterministic kernel. Let m be an odd integer. Under the assumption that the quadratic variation [M] of M is differentiable with E[|d[M](t)/dt|^m] finite, it is shown that the mth power variation [EQUATION PRESANTED] exists and is zero when a quantity δ²(r) related to the variance of an increment of M over a small interval of length r satisfies δ(r) = o(r^1/(2m)). When M is the Wiener process, X is Gaussian; the class then includes fractional Brownian motion and other Gaussian processes with or without stationary increments. When X is Gaussian and has stationary increments, δ is X's univariate canonical metric, and the condition on δ is proved to be necessary. In the non-stationary Gaussian case, when m = 3, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô's formula is established for all functions of class C⁶.