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 $\mathbf{E}\left[ \left\vert d\left[ M\right] (t)/dt\right\vert ^{m}\right]$ finite, it is shown that the $m$th power variation $\lim_{\varepsilon\rightarrow0}\varepsilon^{-1}\int_{0}^{T}ds\left( X\left( s+\varepsilon\right) -X\left( s\right) \right) ^{m}$ exists and is zero when a quantity $δ^{2}\left( r\right)$ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $δ\left( r\right) =o\left( r^{1/(2m)}\right)$. 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, $δ$ 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 formula is established for all functions of class $C^{6}$.

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