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

Abstract: Let ${X_{1}(t)}<em>{0\leq t\leq1}$ and ${X</em>{2}(t)}<em>{0\leq t\leq1}$ be two independent continuous centered Gaussian processes with covariance functions$R</em>{1}$ and $R_{2}$. This paper shows that if the covariance functions are of finite $p$-variation and $q$-variation respectively and such that $p<sup>{-1}+q<sup>{-1}&gt;1$,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.