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 - <p style="text-align: justify;">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.</p>