On some transformations between positive self--similar Markov processes

Loïc Chaumont (1) , Víctor Manuel Rivero (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 CIMAT - Centro de Investigación en Matemáticas

A path decomposition at the infimum for positive self-similar Markov processes (pssMp) is obtained. Next, several aspects of the conditioning to hit 0 of a pssMp are studied. Associated to a given a pssMp $X,$ that never hits 0, we construct a pssMp $X^{\downarrow}$ that hits 0 in a finite time. The latter can be viewed as $X$ conditioned to hit 0 in a finite time and we prove that this conditioning is determined by the pre-minimum part of $X.$ Finally, we provide a method for conditioning a pssMp that hits 0 by a jump to do it continuously.

Mots clés: Self-similar Markov processes; Lévy processes; weak convergence; decomposition at the minimum; conditioning; $h$-transforms

Domaines: Probabilités [math.PR]

Loïc Chaumont, Víctor Manuel Rivero. On some transformations between positive self--similar Markov processes. 2006. ⟨hal-00016781⟩