The Erlang(n) risk model with two-sided jumps and a constant dividend barrier

Abstract

In this paper, the Erlang(n) risk model with two-sided jumps and a constant dividend barrier is considered. In the analysis of the expected discounted penalty function, the downward jumps are assumed to have an arbitrary distribution function and the upward jumps are assumed to be exponentially distributed. An integro-differential equation with boundary conditions for the expected discounted penalty function is derived and the solution is provided. The defective renewal equation for the expected discounted penalty function with no barrier is derived. In the analysis of the moments of the discounted dividend payments until ruin, we assume that the inter-jump times are generalized Erlang(n) distributed. An integro-differential equation for the mth moment function of the discounted sum of dividend payments until ruin is derived. Numerical examples are also given to obtain the expressions for the expected discounted penalty function and the expected present value of dividend payments.