Parisian ruin in a discrete-time Markov-modulated dual risk model

Abstract

In this paper, we investigate the Parisian ruin problems in a discrete-time Markov-modulated dual risk model, wherein the gain process is governed by the underlying Markov process with a finite state space. By using the strong Markov property of the risk process, we derive recursive expressions for the conditional probability generating functions of the classical ruin time and the Parisian ruin time. From this, we not only obtain the infinite-time ruin probabilities but also compute the finite-time ruin probabilities by using numerical inversion. In addition, for the case in which the gain amounts have discrete phase-type distributions, we obtain specialized expressions for the probability generating functions of the classical and Parisian ruin times, which can be used to reduce the computational effort needed for the numerical computation of the ruin probabilities. Finally, we present numerical examples for the computation of the finite- and infinite-time ruin probabilities.