Tail bounds for the joint distribution of the surplus prior to and at ruin

For the classical risk model with Poisson arrivals, we study the (bivariate) tail of the joint distribution of the surplus prior to and at ruin. We obtain some exact expressions and new bounds for this tail, and we suggest three numerical methods that may yield upper and lower bounds for it. As a by-product of the analysis, we obtain new upper and lower bounds for the probability and severity of ruin. Many of the bounds in the present paper improve and generalise corresponding bounds that have appeared earlier. For the numerical bounds, their performance is also compared against bounds available in the literature.