  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.
