Dividend Barrier Strategies in a Renewal Risk Model With Phase-Type Distributed Interclaim Times

In this paper, we consider the optimal dividend problem of 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. We study the optimal dividend under the 2--order and the $n$--order ($n\ge 3$)separately. In the case of 2--order phase-type distributed interclaim times, we show that the optimal dividend policy is the optimal phase-wise barrier strategy. As a byproduct, we find that in this case, the phase with higher barrier is the phase with the higher intensity to the next claim. In the case of n-order ($n\ge3$) phase-type distributed interclaim times, an iteration algorithm is presented to show that the optimal phase-wise barrier strategy is optimal among all dividend policies. We also find a similar conclusion like in the case of 2--order, the phase with the highest barrier is the phase with the highest intensity to the next claim.