Proof(s) of the Lamperti representation of Continuous-State Branching Processes

Maria-Emilia Caballero (1) , Amaury Lambert (2) , Geronimo Uribe Bravo (1)

1 Instituto de Matematicas [México]
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

This paper uses two new ingredients, namely stochastic differential equations satisfied by continuous-state branching processes (CSBPs), and a topology under which the Lamperti transformation is continuous, in order to provide self-contained proofs of Lamperti's 1967 representation of CSBPs in terms of spectrally positive Lévy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is evident.

Mots clés: Continuous-state branching processes; spectrally positive Lévy processes; random time change; stochastic integral equations; Skorohod topology

Domaines: Probabilités [math.PR]

Maria-Emilia Caballero, Amaury Lambert, Geronimo Uribe Bravo. Proof(s) of the Lamperti representation of Continuous-State Branching Processes. 2009. ⟨hal-00257619v2⟩