On the Discounted Penalty at Ruin in the Erlang (2) Risk Process

Abstract

Recently actuarial science practitioners have paid increasing attention to the expected discounted penalty at ruin. From a mathematical point of view, the joint distribution of three random variables, the deficit at ruin, the time at ruin, and the surplus immediately before ruin, is embedded in the unified study of an expected discounted penalty. 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 integral-differential equation, and give its initial value, i.e., its value at zero initial surplus, as well as its Laplace transform. We further show that this function satisfies a Volterra-type integral equation, and a explicit expression of this function can be derived as the solution to this integral equation. As a special case of the expected discounted penalty function, some results for the discounted joint and marginal distributions of the surplus immediately prior to the ruin and the deficit at the time of ruin, and for the discounted distribution function of the amount of the claim causing ruin, as well as the moment generating function of the time of ruin and the ruin probability are also obtained.