The joint distribution of the surplus prior to ruin and the deficit at ruin in some Sparre Andersen models

For the Sparre Andersen risk model, we derive a general expression for h(u,x,y), the joint density function of the surplus prior to ruin and the deficit at ruin when the initial surplus is u. This density function is expressed in terms of the corresponding density function when the initial surplus is 0. We apply a known result for h(0,x,y) in the situation when inter-claim times follow a generalised Erlang distribution to derive expressions for h(u,x,y) when individual claims have a phase-type(m) distribution, m∈Z+. We also consider the case when inter-claim times follow a phase-type(2) distribution and derive an expression for h(0,x,y).