Functional limit theorems for Lévy processes satisfying Cramer's condition
Résumé
We consider a Levy process that starts from x < 0 and conditioned on having a positive maximum. When Cramer's condition holds, we provide two weak limit theorems as x -> -infinity for the law of the (two-sided) path shifted at the first instant when it enters (0, infinity), respectively shifted at the instant when its overall maximum is reached. The comparison of these two asymptotic results yields some interesting identities related to time-reversal, insurance risk, and self-similar Markov processes