Semimartingale dynamics and estimation for a semi-Markov chain

TL;DR

This paper develops a mathematical framework for semi-Markov chains, deriving their semimartingale dynamics, and introduces estimation techniques for occupation times and filtering in noisy observations.

Contribution

It extends the theory of Markov chains to semi-Markov processes, providing new dynamics and estimation methods for noisy semi-Markov observations.

Findings

Derived semimartingale dynamics for semi-Markov chains

Proposed estimators for occupation times in states

Developed filters and smoothers for noisy observations

Abstract

We consider a finite state discrete time process X. Without loss of generality the finite state space can be identified with the set of unit vectors {e1, e2, . . . , eN} with ei = (0, . . . , 0, 1, 0, . . . , 0)0 2 RN. For a Markov chain the times the process stays in any state are geometrically distributed. This condition is relaxed for a semi-Markov chain. We first derive the semimartingale dynamics for a semi-Markov chain. We then consider the situation where the chain is observed in noise. We suggest how to estimate the occupation times in the states and derive filters and smoothers for quantities associated with the chain.