On Filtering Equations in Infinite Dimensions

We consider a nonlinear filtering problem for a state processXin a Hilbert spaceH, given a finite dimensional observation processY. We look for solutions of the corresponding filtering equations, having densities with respect to a Gaussian reference measure inH. We derive the equations for the conditional densities and prove that they have solutions. We can allow correlated noise in the state and observation equations, provided additional assumptions hold.