On the Distribution of the Deficit at Ruin when Claims are Phase-type

We consider the distribution of the deficit at ruin in the Sparre Andersen renewal risk model given that ruin occurs. We show that if the individual claim amounts have a phase-type distribution, then there is a simple phase-type representation for the distribution of the deficit. We illustrate the application of this result with several examples. Keywords: Phase-type Distribution, Sparre Andersen Model, Renewal Risk Model, Deficit At Ruin, Ladder Height, Maximal Aggregate Loss, Erlang Distribution, Coxian Distribution