TY - JOUR T1 - A Joint Density Function in the Renewal Risk Model AU - Xu , Huai AU - Tang , Ling JO - Communications in Mathematical Research VL - 1 SP - 88 EP - 96 PY - 2021 DA - 2021/05 SN - 29 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19031.html KW - deficit at ruin, surplus prior to ruin, phase-type distribution, renewal risk model, maximal aggregate loss. AB -

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.