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 Lévy 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.
"A Lamperti-type representation of continuous-state branching processes with immigration." Ann. Probab. 41 (3A) 1585 - 1627, May 2013. https://doi.org/10.1214/12-AOP766