The expected discounted penalty at ruin in the Erlang (2) risk process

In this paper, under the Erlang (2) risk process, we examine the expected discounted value of a penalty at ruin, which is considered as a function of the initial surplus. We first show that the expected discounted penalty function satisfies an integro-differential equation, and give its initial value, as well as its Laplace transform. We further prove that this function is twice differentiable, and satisfies a defective renewal equation. An explicit expression for the solution of this equation can be derived. The associated compound geometric distribution and “claim size” distribution are also studied.