  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), (for t nonnegative).
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 and show that this 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 the
natural filtration of Y.
