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

TL;DR

This paper offers new proofs of Lamperti's 1967 representation of continuous-state branching processes using stochastic differential equations and a specialized topology, enhancing understanding of their connection to spectrally positive Lévy processes.

Contribution

It provides two self-contained proofs of Lamperti's representation, one probabilistic and one via discrete approximations, utilizing novel ingredients like SDEs and a specific topology.

Findings

Probabilistic proof of Lamperti's representation

Approximation-based proof using discrete processes

Enhanced understanding of CSBPs and Lévy processes

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.