Semimartingle Representation of a class of Semi-Markov Dynamics

TL;DR

This paper develops a semimartingale representation for a class of semi-Markov processes with non-homogeneous embedded chains, establishing existence, uniqueness, and properties of the solutions.

Contribution

It introduces a stochastic integral equation representation for semi-Markov processes with non-homogeneous embedded chains, proving existence, uniqueness, and deriving their law.

Findings

Representation as semi-martingales using Poisson random measures

Existence and uniqueness of the stochastic integral equation

Law of the bivariate process from different initial conditions

Abstract

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 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.