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

TL;DR

This paper studies the power variation of a broad class of stochastic processes, including Gaussian and non-Gaussian types, establishing conditions for their variation to vanish and developing related stochastic calculus tools.

Contribution

It introduces new conditions under which the $m$th power variation of these processes exists and is zero, and extends stochastic calculus to non-stationary Gaussian processes.

Findings

Power variation exists and is zero under certain smoothness conditions.

Includes Gaussian processes like fractional Brownian motion within the framework.

Defines and proves the existence of symmetric integrals for non-stationary Gaussian processes.

Abstract

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 $\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) $. 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 It\^o formula is established for all functions of class $C^{6}$.