Overshoots and undershoots of Lévy processes

We obtain a new fluctuation identity for a general Lévy process giv-ing a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Klüppelberg et al. (2004) concerning asymptotic overshoot distribution of a particular class of Lévy processes with semi-heavy tails and refine some of their main conclusions. In particular we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applica-tions in insurance mathematics are noted with emphasis on the case that the underlying Lévy process is spectrally one sided. 1. Lévy processes and ladder processes This paper concerns overshoots and undershoots of Lévy processes at first upwards passage of a constant boundary. We will therefore begin by introducing some necessary but standard notation. In the sequel X will always denote a Lévy process defined on the filtered spac