The Gerber–Shiu penalty functions for two classes of renewal risk processes

In this paper, we study the Gerber–Shiu functions for a risk model with two independent classes of risks. We suppose that both of the two claim number processes are renewal processes with phase-type inter-claim times. By re-composing and analyzing the Markov chains associated with two given phase-type distributions, we obtain systems of integro-differential equations for two types of Gerber–Shiu functions. Explicit expressions for the Laplace transforms of the two types of Gerber–Shiu functions are established, respectively. And explicit results for the Gerber–Shiu functions are derived when the initial surplus is zero and when the two claim amount distributions are both from the rational family. Finally, an example is considered to illustrate the applicability of our main results.