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

Let {X1(t)}0≤t≤1{X1(t)}0≤t≤1 and {X2(t)}0≤t≤1{X2(t)}0≤t≤1 be two independent continuous centered Gaussian processes with covariance functions R1R1 and R2R2. We show that if the covariance functions are of finite pp-variation and qq-variation respectively and such that p−1+q−1>1p−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 X1X1 and X2X2. Moreover, some properties of the characteristic function of that generalised Lévy area are studied.