On Lévy processes conditioned to stay positive.

Loïc Chaumont (1) , Ron Doney (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Department of Mathematics

We construct the law of Lévy processes conditioned to stay positive under general hypotheses. We obtain a Williams type path decomposition at the minimum of these processes. This result is then applied to prove the weak convergence of the law of Lévy processes conditioned to stay positive as their initial state tends to 0. We describe an absolute continuity relationship between the limit law and the measure of the excursions away from 0 of the underlying Lévy process reflected at its minimum. Then, when the Lévy process creeps upwards, we study the lower tail at 0 of the law of the height this excursion.

Mots clés: Lévy process conditioned to stay positive; path decomposition; weak convergence; excursion measure; creeping.; creeping

Domaines: Probabilités [math.PR]

Loïc Chaumont, Ron Doney. On Lévy processes conditioned to stay positive.. 2005. ⟨hal-00003128v2⟩