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

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 exponentially distributed.