Weak convergence approach to compound Poisson risk processes perturbed by diffusion

We obtain the ruin probability and expected discounted penalty function for a diffusion-perturbed classical risk model, by taking limits in a sequence of compound Poisson processes that converge weakly to the former. This allows us to improve upon a result of Tsai and Willmot [Tsai, C.C.L., Willmot, G.E., 2002. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance Math. Econ. 30, 51–66].