Gaussian and non-Gaussian processes of zero power variation, and related stochastic calculus

We consider a class of stochastic processes $X$ defined by $X\left( t\right) =\int_{0}^{T}G\left( t,s\right) dM\left( s\right) $for$t\in\lbrack0,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 $\mathbf{E}\left[{\mid d\left[M\right](t)\mathrm{/}dt\mid }^m\right]$ finite, it is shown that the $m$th power variation $$\underset{\epsilon \to 0}{\lim }{\epsilon }^{-1}{\int }_0^{T}ds{(X(s+\epsilon )-X\left(s\right))}^m$$ 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/(2m)}\right) $\mathrm{.}When$M$istheWienerprocess,$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 It^o formula is established for all functions of class $C^6$.