(untitled) (Communications in Mathematical Research, 2021)
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.