A Joint Density Function in the Renewal Risk Model

Abstract

In this paper, we consider a general expression for $ϕ(u, x, y)$, the joint density function of the surplus prior to ruin and the deficit at ruin when the initial surplus is $u$. In the renewal risk model, this density function is expressed in terms of the corresponding density function when the initial surplus is 0. In the compound Poisson risk process with phase-type claim size, we derive an explicit expression for $ϕ(u, x, y)$. Finally, we give a numerical example to illustrate the application of these results.