A Support Theorem for the Filter under Infinite Dimensional Noise and Unbounded Observation Coefficients

In this paper we consider a nonlinear filtering problem with an unbounded observation coefficient, correlated noises, and a signal process driven by an infinite dimensional Brownian motion. We prove that the unnormalized filter admits a smooth density which is in the Schwartz space and we give a description of the support of the law of this density. Accepted 1 December 1997