Conditioned stable Lévy processes and Lamperti representation

Abstract

By killing a stable Lévy process when it leaves the positive half-line, or by conditioning it to stay positive, or by conditioning it to hit 0 continuously, we obtain three different positive self-similar Markov processes which illustrate the three classes described by Lamperti \cite{La}. For each of these processes, we compute explicitly the infinitesimal generator from which we deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0 which we describe here. As an application, we give the law of the minimum before an independent exponential time of a certain class of Lévy processes. It provides the explicit form of the spacial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.