On a class of renewal risk models with a constant dividend barrier

Abstract

We consider a compound renewal (Sparre Andersen) risk process in the presence of a constant dividend barrier in which the claim waiting times are generalized Erlang(n) distributed (i.e., convolution of n exponential distributions with possibly different parameters). An integro-differential equation with certain boundary conditions for the Gerber-Shiu function is derived and solved. Its solution can be expressed as the Gerber-Shiu function in the corresponding Sparre Andersen risk model without a barrier plus a linear combination of n linearly independent solutions to the associated homogeneous integro-differential equation. Finally, explicit results are given when the claim sizes are exponen..