The Gerber-Shiu Function for a Sparre Andersen\\Risk Model

In this paper, we consider a Sparre Andersen risk model in which the claim inter-arrival distribution is a mixture of an Erlang(n) distribution and an Erlang(m) distribution. Our purpose is to study the Gerber-Shiu function \phi_\delta(u). First, we show that \phi_\delta(u) satisfies a higher-order integro-differential equation, and then we discuss the roots of the generalized Lundberg's equation. On this basis, we drive the Laplace transform of \phi_\delta(u) and prove that \phi_\delta(u) satisfies a renewal equation. Finally, we consider an example.