Expected discounted penalty function and asymptotic depend of the severity of ruin and surplus prior to ruin for two-sided Levy risk processes

We study a generalization of the Expected Discounted Penalty Function (EDPF) for a class of two-sided jump Levy processes R. having positive jumps with a rational Laplace transform. Our first result provides an explicit expression for the generalized EDPF in terms of functions depending only on the parameters of the Levy process R. Later on, we apply our results to study a measure of asymptotic dependence for the severity of ruin on the surplus prior to ruin, for the class of Levy risk processes considered in this work.