Proof(s) of the Lamperti representation of continuous-state branching processes. (English) Zbl 1194.60053

Summary: 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 J. Lamperti’s [Bull. Am. Math. Soc. 73, 382–386 (1967; Zbl 0173.20103)] 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.


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


Zbl 0173.20103
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