A Joint Density Function in the Renewal Risk Model (Communications in Mathematical Research, 2021)
In this paper, we consider a general expression for $ϕ(u, x, y)$, the jointdensity function of the surplus prior to ruin and the deficit at ruin when the initialsurplus is $u$. In the renewal risk model, this density function is expressed in termsof the corresponding density function when the initial surplus is 0. In the compoundPoisson 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 theseresults.
Keywords: deficit at ruin; surplus prior to ruin; phase-type distribution; renewal risk model; maximal aggregate loss.