Optimal Dividend Strategies in a Renewal Risk Model with Phase-Type Distributed Interclaim Times

In this paper, we consider the optimal dividend problem for the renewal risk model with phase-type distributed interclaim times and exponentially distributed claim sizes. Assume that the phases of the interclaim times can be observed. The goal is to find the optimal dividend policy to maximize the cumulative discounted dividend before ruin. To explore the optimal strategy, we first present an algorithm and then for each particular phase-type distributed interclaim times example, the optimality of phase-wise barrier strategy as well as the convergence of the algorithm is proved. Then we theoretically analyze some properties of the value function and the optimal phase-wise barriers should fulfill. Furthermore, we specifically analyze the concavity of the value function and the barrier size comparison in the case of 2-order distributed interclaim times. In the last, we theoretically show that the phase with the highest barrier is the one with the highest intensity to the next claim.