Occupation densities for stochastic integral processes in the second Wiener chaos

Summary The two-point distributions of Skorohod integral processes in the second Wiener chaos are mainly described by a Hilbert-Schmidt operatorT giving the mutual interaction of infinitely many Gaussian components and by simple multiplication operators. So are the Fourier transforms of their occupation measures. This enables us to use the well known Fourier analytic criterion discovered and elaborated by Berman to derive integral conditions for the existence of their occupation densities in terms of associated Hilbert-Schmidt operators. IfT is a trace class operator, we get a necessary and sufficient criterion, if it is not, still a sufficient one. In a case in which the interaction is particularly simple, we verify the appropriate integral condition and show that the results are essentially beyond the reach of enlargement of filtrations techniques of semimartingale theory.