  An approximation to the solution of a stochastic parabolic equation is
constructed using the Galerkin approximation followed by the Wiener Chaos
decomposition. The result is applied to the nonlinear filtering problem for the
time homogeneous diffusion model with correlated noise. An algorithm is
proposed for computing recursive approximations of the unnormalized filtering
density and filter, and the errors of the approximations are estimated. Unlike
most existing algorithms for nonlinear filtering, the real-time part of the
algorithm does not require solving partial differential equations or evaluating
integrals. The algorithm can be used for both continuous and discrete time
observations.
