Semimartingale Representation of a Class of Semi-Markov Dynamics

We consider a class of semi-Markov processes (SMP) such that the embedded discrete-time Markov chain may be non-homogeneous. The corresponding augmented processes are represented as semi-martingales using a stochastic integral equation involving a Poisson random measure. The existence and uniqueness of the equation are established. Subsequently, we show that the solution is indeed a SMP with desired transition rate. Finally, we derive the law of the bivariate process obtained from two solutions of the equation having two different initial conditions.