A note on limiting distribution for jumps of Lévy insurance risk model

Results of Doney and Kyprianou (2006) and Park and Maller (2008) for asymptotic overshoot and undershoot distributions in the class of a general Lévy process with convolution equivalent measures are used to obtain the limiting distribution of the maximum process before ruin, the undershoot immediately before ruin and the overshoot at ruin, associated with the ruin time. It allows us to study estimation and derivation of limiting distribution extensions for jumps when a ruin occurs. Numerical study of finite sample versions are given for specific illustrations of the limiting distributions.