Non-Anticipative Representations of Equivalent Gaussian Processes

Abstract

Given two equivalent Gaussian processes the notion of a non-anticipative representation of one of the processes with respect to the other is defined. The main theorem establishes the existence of such a representation under very general conditions. The result is applied to derive such representations explicitly in two important cases where one of the processes is (i) a Wiener process, and (ii) a $N$-ple Gaussian Markov process. Radon-Nikodym derivatives are also discussed.