The Semimartingale Dynamics and Generator of a Continuous Time Semi-Markov Chain

We consider a finite state, continuous time homogeneous semiMarkov chain X = {Xt, t ≥ 0}. Without loss of generality the state space of the chain can be identified with the set of unit vectors S = {e1, e2, . . . , eN} where ei = (0, . . . , 0, 1, 0, . . . , 0) ′ ∈ RN . The probabilistic and dynamic properties of X can be described by either a rate matrix A or a matrix which gives the occupation times in the various states together with the probabilities of jumping to a different state. For a continuous time Markov chain the occupation times are memoryless, implying the distributions are exponential. For semi-Markov chains the occupation times can have more general distributions. The relation between these two descriptions is first investigated and the semimartingale dynamics of a semi-Markov chain obtained in contrast to the traditional description of a semi-Markov chain in terms of a renewal process. An equation giving the dynamics of the occupation times is derived together with an equation for the density of the conditional occupation time and state. Some approximations for these dynamics are then obtained.