Spectral Representation of Gaussian Semimartingales



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 ℱ X -semimartingale property is considered, and afterwards the ℱ X,∞-semimartingale property is treated in the case where X is a moving average process and ℱ t X,∞ =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 ℱ W,∞-semimartingale (W denotes a Wiener process). Hereby we generalize a result of Knight (Foundations of the Prediction Process, 1992) to the nonstationary case.


Semimartingales Gaussian processes Volterra processes Stationary processes Moving average processes 

Mathematics Subject Classification (2000)

60G15 60G10 60G48 60G57 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of AarhusÅrhus CDenmark

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