Gaussian and non-Gaussian processes of zero power variation

¹ ENSTA-ParisTech. Unité de Mathématiques appliquées, 828, bd des Maréchaux, 91120 Palaiseau, France
² INRIA Rocquencourt, Projet MathFi and Cermics, École des Ponts, Rocquencourt, France
³ Department of Statistics, Purdue University, 150 N. University St., West Lafayette, IN 47907-2067, USA
viens@stat.purdue.edu

Received: 9 August 2012
Revised: 21 July 2014

Abstract

We consider a class of stochastic processes X defined by X(t) = ∫₀^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 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.