GAUSSIAN AND NON-GAUSSIAN PROCESSES OF ZERO POWER VARIATION

We consider a class of stochastic processes X defined by X (t) = integral(T)(0) G (t, s) dM (s) for t is an element of [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 [vertical bar d [M] (t)/dt vertical bar(m)] finite, it is shown that the mth power variation lim(epsilon -> 0) epsilon(-1) integral(T)(0) ds (X (s + epsilon) - X (s))(m) exists and is zero when a quantity delta(2) (r) related to the variance of an increment of M over a small interval of length r satisfies delta(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, delta is X's univariate canonical metric, and the condition on delta 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 Ito's formula is established for all functions of class C-6.