Gaussian and non-Gaussian processes of zero power variation

Russo, Francesco; Viens, Frederi

ESAIM: Probability and Statistics, Tome 19 (2015), pp. 414-439.

We consider a class of stochastic processes $X$ defined by $X\left(t\right)=\int {}_0^T$ $G(t,s)dM\left(s\right)$ for $t\in [0,T]$, 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 $\left[M\right]$ of $M$ is differentiable with $𝐄\left[\right|d\left[M\right]\left(t\right)/dt{|}^m]$ finite, it is shown that the $m$th power variation

$$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim}{\epsilon }^{-1}{\int }_0^T\mathrm{d}s{\left(X\left(s+\epsilon \right)-X\left(s\right)\right)}^m\end{array}$$

exists and is zero when a quantity $\delta {}^2\left(r\right)$ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta \left(r\right)=o\left(r^{1/\left(2m\right)}\right)$. 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 Itô’s formula is established for all functions of class $C^6$.

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