A new fluctuation identity for Lévy processes and some applications

Abstract

Let τ and H be respectively the ladder time and ladder height processes associated with a given Lévy process X. We give an identity in law between (τ,H) and (X,H^*), H^* being the right-continuous inverse of the process H. This allows us to obtain a relationship between the entrance law of X and the entrance law of the excursion measure away from 0 of the reflected process (X_t- inf_s≤tX_s, t ≥0). In the stable case, some explicit calculations are provided. These results also lead to an explicit form of the entrance law of the Lévy process conditioned to stay positive.