Existence of a density for the filter associated to Hilbert-space valued systems

The purpose of this paper is to prove that the unnormalized filter associated to a Hilbert space valued nonlinear filtering problem with correlated noises, which solves a parabolic stochastic partial differential equation, the Zakai equation, admits a density with respect to a given measure on the Hilbert space. We also compute the stochastic differential equation solved by this density.