The expected discounted penalty function: from infinite time to finite time

In this paper we study the finite-time expected discounted penalty function (EDPF) and its decomposition in the classical risk model perturbed by diffusion. We first give the solution to a class of second-order partial integro-differential equations (PIDEs) with certain boundary conditions. We then show that the finite-time EDPFs as well as their decompositions satisfy this specific class of PIDEs so that their explicit expressions are obtained. Furthermore, we demonstrate that the finite-time EDPF may be expressed in terms of its ordinary counterpart (infinite-time) under the same risk model. Especially, the finite-time ruin probability due to oscillations and the finite-time ruin probability caused by a claim may also be expressed in terms of the corresponding quantities under the infinite-time horizon. Numerical examples are given when claims follow an exponential distribution.