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

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

2010-09-06

arxiv.org/abs/1009.1039v1

Mathematics

Probability

Scientific paper

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.

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