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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. [ABSTRACT FROM AUTHOR]

Abstract: This note focuses on the Ornstein–Uhlenbeck process reflected at its long-run level (or long-run mean). The analytical closed-form of the transition density is obtained by virtue of the Skorokhod equation and the time-change for martingales. Our result is consistent with that presented by . Finally, an open problem concerning the general cases (reflected at an arbitrary level) is proposed. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a jump diffusion risk model, which consists of a Brownian motion, phase type distributed positive claims and general negative claims. The distributions of the time to ruin and the deficit at ruin will be studied by using Rouché’s Theorem, martingale and matrix analysis. We derive an explicit joint Laplace transform for the time to ruin and the deficit at ruin, as well as the Laplace transform for the time to ruin. Furthermore, our results still hold even when positive claims are rationally distributed. [Copyright &y& Elsevier]