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Stochastic Models for the Inference of Life Evolution

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

Caballero, M., Lambert, A., Uribe Bravo, G.

Probab. Surveys

2009

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 L\'evy 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.

Bibtex

@article{caballero_proofs_2009,
Author = {Caballero, Maria-Emilia and Lambert, Amaury and Uribe
Bravo, Geronimo},
Title = {Proof(s) of the {Lamperti} representation of
continuous-state branching processes},
Journal = {Probab. Surveys},
Volume = {6},
Pages = {62--89},
abstract = {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
L\'evy 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. },
doi = {10.1214/09-PS154},
year = 2009
}

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