Semimartingale dynamics and estimation for a semi-Markov chain

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.