The Gerber–Shiu discounted penalty function in the risk process with phase-type interclaim times

In this paper, we consider the Gerber–Shiu discounted penalty function for the Sparre Anderson risk process in which the interclaim times have a phase-type distribution. By the Markov property of a joint process composed of the risk process and the underlying Markov process, we provide a new approach to prove the systems of integro-differential equations for the Gerber–Shiu functions. Closed form expressions for the Gerber–Shiu functions are obtained when the claim amount distribution is from the rational family. Finally we compute several numerical examples intended to illustrate the main results.