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

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 ), (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 , (), where 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 .

Keywords:

• nonlinear filtering
• piecewise-deterministic Markov processes
• optimal stopping

Keywords:

• 60G35
• 93E11
• 60G40

Notes

^1. [email protected]

Additional information

Notes on contributors

Fulvia Confortola

1 1. [email protected]