Gaussian and non-Gaussian processes of zero power variation

TL;DR

This paper investigates the power variation of stochastic processes formed by Volterra convolutions of martingales, establishing conditions under which these variations are zero, and extends results to non-Gaussian and non-homogeneous Gaussian processes.

Contribution

It provides new conditions for zero power variation in Volterra processes, including non-Gaussian cases, and develops a generalized Stratonovich integral with Itô's formula.

Findings

Power variation exists and is zero under specific conditions on the quadratic variation.

Necessary and sufficient conditions are established for Gaussian processes with homogeneous increments.

A generalized Stratonovich integral is defined and its Itô's formula proved for certain non-homogeneous Gaussian processes.

Abstract

This paper considers the class of stochastic processes $X$ which are Volterra convolutions of a martingale $M$. When $M$ is Brownian motion, $X$ is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let $m$ be an odd integer. Under some technical conditions on the quadratic variation of $M$, it is shown that the $m$-power variation exists and is zero when a quantity $\delta^{2}(r) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta(r) = o(r^{1/(2m)}) $. In the case of a Gaussian process with homogeneous increments, $\delta$ is $X$'s canonical metric and the condition on $\delta$ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers. In the non-homogeneous Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its It\^o's formula is proved to hold for all functions of class $C^{6}$.