Filtering with discrete state observations

The problem of estimating a finite state Markov chain observed via a process on the same state space is discussed. Optimal solutions are given for both the `weak´ and `strong´ formulations of the problem. The `weak´ formulation proceeds using a reference probability and a measure change for Markov chains. The `strong´ formulation considers an observation process related to perturbations of the counting processes associated with the Markov chain. In this case the `small noise´ convergence is investigated