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

TL;DR

This paper develops explicit filtering equations for continuous-time Markov chains observed through noise-free functions, demonstrating their Markov and piecewise-deterministic properties, and applies this to an optimal stopping problem.

Contribution

It provides explicit filtering equations for noise-free observations of Markov chains and characterizes the resulting process as a piecewise-deterministic Markov process.

Findings

Explicit filtering equations derived for noise-free observations.

The filtering process is shown to be a Markov and Feller process.

Application to optimal stopping with partial observation.

Abstract

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.