Sample path behavior of a Lévy insurance risk process approaching ruin, under the Cramér-Lundberg and convolution equivalent conditions

Recent studies have demonstrated an interesting connection between the asymptotic behavior at ruin of a Lévy insurance risk process under the Cramér-Lundberg and convolution equivalent conditions. For example, the limiting distributions of the overshoot and the undershoot are strikingly similar in these two settings. This is somewhat surprising since the global sample path behavior of the process under these two conditions is quite different. Using tools from excursion theory and fluctuation theory, we provide a means of transferring results from one setting to the other which, among other things, explains this connection and leads to new asymptotic results. This is done by describing the evolution of the sample paths from the time of the last maximum prior to ruin until ruin occurs.