Probability Surveys

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

Ma. Emilia Caballero, Amaury Lambert, and Gerónimo Uribe Bravo

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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.

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Probab. Surveys, Volume 6 (2009), 62-89.

First available in Project Euclid: 1 December 2009

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60B10: Convergence of probability measures 60G44: Martingales with continuous parameter 60G51: Processes with independent increments; Lévy processes 60H20: Stochastic integral equations

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


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

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