On continuous state branching processes: conditioning and self-similarity

TL;DR

This paper demonstrates that certain stable continuous-state branching processes and their conditioned versions are positive self-similar Markov processes, providing explicit path results through Lamperti transformations.

Contribution

It establishes the self-similarity of conditioned stable continuous-state branching processes and explores their path properties via Lamperti transformations.

Findings

Stable CSB processes are positive self-similar Markov processes.

Explicit path descriptions for conditioned processes are derived.

Connections between different Lamperti transformations are clarified.

Abstract

In this paper, for $\alpha\in (1, 2}$ we show that the $\alpha$-stable continuous-state branching process and the associated process conditioned never to become extinct are positive self-similar Markov processes. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive self-similar Markov processes permits accessto a number of explicit results concerning the paths of stable-continuous branching processes and its conditioned version.