Let X be a continuous-time Markov chain in a finite set I, let h be a mapping of I onto another set, and let Y be defined by Y t = h(X t ), (t ≥ 0). We address the filtering problem for X in terms of the observation Y , which is not directly affected by noise. We write down explicit equations for the filtering process Π tWe show that Π is a Markov process with the Feller property. We also prove that it is a piecewise-deterministic Markov process in the sense of Davis, and we identify its characteristics explicitly. We finally solve an optimal stopping problem for X with partial observation, i.e. where the moment of stopping is required to be a stopping time with respect to (Y 0 t ).Recently, the following different model has been addressed by several authors: