Filtering with discrete state observations

The problem of estimating a finite state Markov chain observed via a proces s on the same state space is discussed. Optimal solutions are given for bot h the "weak" and "strong" formulations of the problem. The "weak" formulati on proceeds using a reference probability and a measure change for the Mark ov chain. The "strong" formulation considers an observation process related to perturbations of the counting processes associated with the Markov chai n. In this case the "small noise" convergence is investigated.