Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases

► Reviews use of a Lévy process for the insurance risk process, including both Cramér and convolution equivalent cases. ► Derives asymptotic results for the overshoot and undershoots under minimal assumptions in the Cramér case. ► Draws attention to a remarkable connection between the Cramér and convolution equivalent results. ► Illustrates this relationship by a numerical comparison when the Lévy process belongs to the GTSC class. ► The analysis suggests a usefully expanded flexibility for modelling the insurance risk process.