Spectral Representation of Gaussian Semimartingales

Abstract

The aim of the present paper is to characterize the spectral representation of Gaussian semimartingales. That is, we provide necessary and sufficient conditions on the kernel K for Xt=∫Kt(s) dNs to be a semimartingale. Here, N denotes an independently scattered Gaussian random measure on a general space S. We study the semimartingale property of X in three different filtrations. First, the ℱX-semimartingale property is considered, and afterwards the ℱX,∞-semimartingale property is treated in the case where X is a moving average process and ℱtX,∞=σ(Xs:s∈(−∞,t]). Finally, we study a generalization of Gaussian Volterra processes. In particular, we provide necessary and sufficient conditions on K for the Gaussian Volterra process ∫−∞tKt(s) dWs to be an ℱW,∞-semimartingale (W denotes a Wiener process). Hereby we generalize a result of Knight (Foundations of the Prediction Process, 1992) to the nonstationary case.