Representation of Gaussian semimartingales with applications to the covariance function

The present paper is concerned with various aspects of Gaussian semimartingales. Firstly, generalizing a result of Stricker, we provide a convenient representation of Gaussian semimartingales X = X(0) + M + A as an F(M) -semimartingale plus a process of bounded variation which is independent of M. Secondly, we study stationary Gaussian semimartingales and their canonical decomposition. Thirdly, we give a new characterization of the covariance function of Gaussian semimartingales, which enable us to characterize the class of martingales and the processes of bounded variation among the Gaussian semimartingales. We conclude with two applications of the results.