Applications of factorization embeddings for Lévy processes

We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes. First, we find the joint distribution of the supremum and the epoch at which it is .attained. if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. Second, we establish general properties of perturbed risk models, and obtain explicit fluctuation identities in the case that the Lévy process is spectrally positive. Third, we study the tail asymptotics for the supremum of a Lévy process under different assumptions on the tail of the Lévy measure.