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

Let {X1(t)}0<=t<=1 and {X2(t)}0<=t<=1 be two independent continuous centered Gaussian processes with covariance functions R1 and R2. We show that if the covariance functions are of finite p-variation and q-variation respectively and such that p-1+q-1>1, then the Lévy area can be defined as a double Wiener-Itô integral with respect to an isonormal Gaussian process induced by X1 and X2. Moreover, some properties of the characteristic function of that generalised Lévy area are studied.