Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration

We first introduce and derive some basic properties of a two-parameters (α, γ ) family of one-sided Lévy processes, with 1 < α < 2 and γ > −α. Their Laplace exponents are given in terms of the Pochhammer symbol as follows ψ )(λ)= c((λ+ γ )α − (γ )α), λ 0, where c is a positive constant, (λ)α = (λ+α) (λ) stands for the Pochhammer symbol and for the Gamma function. These are a generalization of the Brownian motion, since in the limit case α→ 2, we end up to the Laplace exponent of a Brownian motion with drift γ + 2 . Then, we proceed by computing the density of the law of the exponential functional associated to some elements of this family (and their dual) and some transformations of these elements. More precisely, we shall consider the Lévy processes which admit the following Laplace exponent, for any δ > α−1 α , ψ(0,δ)(λ)=ψ(0)(λ)− αδ λ+ α − 1 (0)(λ), λ 0. These densities are expressed in terms of the Wright hypergeometric functions. By means of probabilistic arguments, we derive some interesting properties enjoyed by these functions. On the way, we also characterize explicitly the semi-group of the family of self-similar continuous state branching processes with immigration. © 2008 Elsevier Masson SAS. All rights reserved. E-mail address: [email protected]. 0007-4497/$ – see front matter © 2008 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.bulsci.2008.10.001 356 P. Patie / Bull. Sci. math. 133 (2009) 355–382 Résumé Nous introduisons et étudions quelques propriétés élémentaires d’une famille de processus de Lévy complètement asymmétriques. Leurs lois sont caractérisées par leurs exposants de Laplace qui s’expriment en termes du symbole de Pochhammer. Ensuite, nous calculons la loi de la fonctionnelle exponentielle associée à certains éléments de cette famille et d’une tranformation de ces éléments. Ces lois s’avèrent absolument continues et leurs densités s’expriment en termes des fonctions hypergéométriques de Wright. En utilisant des arguments probabilistes, nous déduisons que ces fonctions possèdent des propriétés analytiques intéressantes. Lors du déroulement de la preuve, nous caractérisons également le semi-groupe des processus auto-similaires de branchement avec immigration. © 2008 Elsevier Masson SAS. All rights reserved. MSC: 60E07; 60G18; 60G51; 33E12