Abstract
In this paper, we consider a continuous-time semi-Markov process (SMP) in Polish spaces. We first introduce the semi-additive functional in semi-Markov cases, a natural generalization of the additive functional of Markov process (MP). The main contribution of this paper is the properties of semi-additive functional aforementioned. First, semi-additive functionals of SMPs are characterized in terms of a càdlàg function with zero initial value and a measurable function. Second, the necessary and sufficient conditions are investigated under which a semi-additive functional of SMP is a semimartingale, a local martingale, or a special semimartingale respectively. By the way, the Itô type formula is given. Finally, we study the expected cumulative discounted value of the semi-additive functional of an SMP. We prove that it solves a measure-valued Poisson equation and give the uniqueness conditions.
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