RESUMEN

The paper discusses two algorithms the Zakai equation in the time-homogeneous diffusion filtering model with possible correlation between the state process and the observation noise. Both algorithms rely on the Cameron- Martin version of the Wiener chaos expansion, so that the approximate filter is a finte linear combination of the chaos elements generated by the observation process. The coefficients in the expansion depended only the deterministic dynamics of the state advances improves the performance of the algorithms in comparison with the most other existing methods of the nonlinear filtering. The paper summarizes the main existing results about these Wiener chaos algorithms and resolves some open questions concerning the convergence of the algorithms in the noise-correlated setting. The presentation includes the necessary background on the Wiener chaos and optimal nonlinear filtering.