Lévy area for Gaussian processes: A double Wiener-Itô integral approach

Abstract

Let $\{X_1(t){\}}_{0\le t\le 1}$ and $\{X_2(t){\}}_{0\le t\le 1}$ 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{é}vy area can be defined as a double Wiener--Itò 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{é}vy area are studied.