Moments of the time of ruin in a renewal risk model with discounted penalty

This paper considers a Sparre Andersen risk process for which the claims inter‐arrival distribution is Generalized Exponential. The purpose of this paper is to find explicit expressions for the moments of time to ruin when a penalty is imposed at ruin.

The study is focused on the function ϕ_δ(u), the expected discounted penalty, which is due at ruin and may depend on the deficit at the time of ruin and also on the surplus prior to ruin. It shows that ϕ_δ(u) satisfies an integro‐differential equation which is solved using Laplace transforms.

The authors have chosen a penalty function, which is independent of the surplus immediately before ruin, and a closed form expression is obtained for ϕ_δ(u), and then solved for the moments of time to ruin.

New results are derived, many of which have mathematical and probabilistic interpretations, and additional insight is gained for the results in the renewal risk model.