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 Probability Surveys > Vol. 6 (2009) open journal systems 

Proof(s) of the Lamperti representation of continuous-state branching processes

Ma. Emilia Caballero, Instituto de Matemáticas, Universidad Nacional Autónoma de México
Amaury Lambert, Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Ma
Gerónimo Uribe Bravo, Departamento de Probabilidad y Estadística, Instituto de Investigaciones en Mat

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.

AMS 2000 subject classifications: Primary 60J80; secondary 60B10, 60G44, 60G51, 60H20.

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

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Caballero, Ma. Emilia, Lambert, Amaury, Bravo, Gerónimo Uribe, Proof(s) of the Lamperti representation of continuous-state branching processes, Probability Surveys, 6, (2009), 62-89 (electronic). DOI: 10.1214/09-PS154.


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