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 X t =∫ K t (s) dN s 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 ℱ X,∞ t =σ(X s :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 ∫ t−∞ K t (s) dW s 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.
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Basse, A. Spectral Representation of Gaussian Semimartingales. J Theor Probab 22, 811–826 (2009). https://doi.org/10.1007/s10959-009-0246-2
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DOI: https://doi.org/10.1007/s10959-009-0246-2