Singular Perturbation and Order Reduction for Filtering Problem.

We consider the problem of optimal filtering of two dimensional diffusion process measured in a noisy channel. We approximate the solution of Zakai equation for the two dimensional process by a solution of Zakai equation for one dimensional process for two models. The first one is fast and slow variables, that is where one element of the process changes much more rapidly than the second one. The second model is the quasi-deterministic case for which the fast element has a small diffusion term. In both cases simple approximated equations for the filtering problem are given that make numerical solution simpler