The Perturbed Sparre Andersen Model with Interest and a Threshold Dividend Strategy

In this paper, we consider a Sparre Andersen model perturbed by diffusion (in which the inter-claim times are generalized Erlang(n)-distributed) with a constant interest under a threshold dividend payment strategy. Under such a strategy, no dividends are paid if the insurer's surplus is below a certain threshold level. When the surplus is above the threshold level, part of the premium income and all of the interest income are paid out as dividends. Integro-differential equations with certain boundary conditions for the moment generating functions and moment functions of the present value of all dividends until ruin are derived. We also derive the integro-differential equations with boundary conditions for the Gerber-Shiu functions. Explicit expressions are given in terms of some functions related to high order integro-differential equations when the inter-claim times are Erlang(2) and Erlang(1) distributed.