Overshoots and undershoots of Levy processes

We obtain a new fluctuation identity for a general Levy process giving 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 Kluppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766-1801] concerning asymptotic overshoot distribution of a particular class of Levy 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. Applications in insurance mathematics are noted with emphasis on the case that the underlying Levy process is spectrally one sided.