Filtering of continuous-time Markov chains with noise-free observation and applications

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, Πt(i) + ℙ(Xt + ipipeY0t),(i ∈ I, t ≥ 0) where (Y0t)is the natural filtration of Y. We 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 (Y0t). © 2013 Copyright Taylor and Francis Group, LLC.