Overshoots and undershoots of Levy processes

We obtain a new identity giving a quintuple law of overshoot, time of

overshoot, undershoot, last maximum, and time of last maximum of a general Levy

process at ?rst passage. The identity is a simple product of the jump measure

and its ascending and descending bivariate renewal measures. With the help of

this identity, we consider applications for passage problems of stable

processes, recovering and extending results of V. Vigon on the bivariate jump

measure of the ascending ladder process of a general Levy process and present

some new results for asymptotic overshoot distributions for Levy processes with

regularly varying jump measures.

(Parts of this talk are based on joint work with Ron Doney and Claudia

Kluppelberg)