Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### A new fluctuation identity for Lévy processes and some applications

Auteur(s):

Code(s) de Classification MSC:

• 60J30 Processes with independent increments
• 60J20 Applications of discrete Markov processes (social mobility, learning theory, industrial processes, etc.), See Also {90B30, 92H10, 92H35, 92J40}

Résumé: Let $\tau$ and $H$ be respectively the ladder time and ladder height processes associated to a given L\'evy process $X$. We give an identity in law between $(\tau,H)$ and $(X,H^*)$, ($H^*$ being the right continuous inverse of the process $H$). The later allows us to get a relationship between the entrance law of $X$ and the entrance law of the excursion measure away from 0 of the reflected process $(X_t- \inf_{s\leq t}X_s\,,\;t\geq0)$. In the stable case, some explicit calculations are provided. These results also lead to an explicit form of the entrance law of the L\'evy process conditioned to stay positive.

Mots Clés: Processus de Lévy ; théorie des fluctuations ; mesure d'excursion

Date: 1999-03-31

Prépublication numéro: PMA-584