Continuous-time approximations for the nonlinear filtering problem

The paper deals with a possible approach to the problem of finite-dimensional filters in the nonlinear case, when the signal is a diffusion process and the observations are corrupted by additive white noise. The approach considers a sequence of finite-dimensional recursive filters that approximate the actual optimal one. The approximating filters are given in terms of functionals of continuous-time Markov chains that converge weakly to the original diffusion. These functionals can be recursively computed via a finite-dimensional Zakai equation, for which the solution is given in terms of a robust input-output relation.