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In this paper, we consider the optimal dividend problem of the renewal riskmodel with phase-type distributed interclaim times and exponentiallydistributed claim sizes. Assume that the phases of the interclaim times can beobserved. 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 interclaimtimes, we show that the optimal dividend policy is the optimal phase-wisebarrier strategy. As a byproduct, we find that in this case, the phase withhigher barrier is the phase with the higher intensity to the next claim. In thecase of n-order ($n\ge3$) phase-type distributed interclaim times, an iterationalgorithm is presented to show that the optimal phase-wise barrier strategy isoptimal among all dividend policies. We also find a similar conclusion like inthe case of 2--order, the phase with the highest barrier is the phase with thehighest intensity to the next claim.

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