A Lamperti-type representation of continuous-state branching processes with immigration

TL;DR

This paper extends the Lamperti representation to include immigration in continuous-state branching processes, providing a new stochastic differential equation framework, stability analysis, and applications to limit theorems and simulation methods.

Contribution

It introduces a novel Lamperti-type representation for processes with immigration using Levy-driven differential equations, expanding theoretical understanding and practical simulation tools.

Findings

Derived a representation using Levy processes and differential equations.

Proved stability and limit theorems for the processes.

Developed a simulation scheme for these processes.

Abstract

Guided by the relationship between the breadth-first walk of a rooted tree and its sequence of generation sizes, we are able to include immigration in the Lamperti representation of continuous-state branching processes. We provide a representation of continuous-state branching processes with immigration by solving a random ordinary differential equation driven by a pair of independent Levy processes. Stability of the solutions is studied and gives, in particular, limit theorems (of a type previously studied by Grimvall, Kawazu and Watanabe and by Li) and a simulation scheme for continuous-state branching processes with immigration. We further apply our stability analysis to extend Pitman's limit theorem concerning Galton-Watson processes conditioned on total population size to more general offspring laws.