Integral representation of random variables with respect to Gaussian processes

TL;DR

This paper extends the integral representation of random variables from fractional Brownian motion to a broader class of Gaussian processes with certain regularity, emphasizing the role of local covariance properties.

Contribution

It generalizes the integral representation result to Gaussian processes with Hölder continuity, highlighting the importance of local covariance structure.

Findings

Representation holds for Gaussian processes with Hölder continuity of order > 1/2

Local covariance properties determine the integral representation

Extends previous results from fractional Brownian motion to wider processes

Abstract

It was shown in Mishura et al. (Stochastic Process. Appl. 123 (2013) 2353-2369), that any random variable can be represented as improper pathwise integral with respect to fractional Brownian motion. In this paper, we extend this result to cover a wide class of Gaussian processes. In particular, we consider a wide class of processes that are H\"{o}lder continuous of order $\alpha>1/2$ and show that only local properties of the covariance function play role for such results.