L\'evy area for Gaussian processes: A double Wiener-It\^o integral approach

TL;DR

This paper establishes conditions under which the Lévy area for two independent Gaussian processes can be represented as a double Wiener-Itô integral, extending the understanding of stochastic integrals for processes with finite p-variation.

Contribution

It introduces a novel representation of the Lévy area as a double Wiener-Itô integral for Gaussian processes with finite p-variation, broadening the scope of stochastic calculus.

Findings

Lévy area can be defined as a double Wiener-Itô integral under certain variation conditions.

Properties of the characteristic function of the generalized Lévy area are analyzed.

Conditions on covariance functions ensure the existence of the Lévy area as a Wiener-Itô integral.

Abstract

Let $\{X_{1}(t)\}_{0\leq t\leq1}$ and $\{X_{2}(t)\}_{0\leq t\leq1}$ be two independent continuous centered Gaussian processes with covariance functions$R_{1}$ and $R_{2}$. This paper shows that if the covariance functions are of finite $p$-variation and $q$-variation respectively and such that $p^{-1}+q^{-1}>1$,then the L{\'e}vy area can be defined as a double Wiener--It\`o integral with respect to an isonormal Gaussian process induced by $X_{1}$ and $X_{2}$. Moreover, some properties of the characteristic function of that generalised L{\'e}vy area are studied.