Self-exciting counting process systems with finite state space

Stochastic systems with counting process output and a finite state space are considered. This leads to studying processes with finite state space that are Markovian with respect to the flow of [sigma]-algebras, that is generated by the counting process. It appears that there is a close relationship between the transition intensities of the Markov process and the intensity of the counting process. Some consequences for a stochastic realization problem are then studied.