Continuous state branching processes in random environment: The Brownian case

TL;DR

This paper studies continuous state branching processes influenced by Brownian motion, analyzing their long-term extinction and explosion behaviors, and providing explicit probabilities and regimes for these phenomena.

Contribution

It introduces a stochastic differential equation framework for these processes and characterizes their asymptotic behaviors and probabilities in various regimes.

Findings

Explicit extinction and explosion probabilities in the stable case

Identification of regimes for asymptotic behaviors

Analysis of conditioned processes and immigration effects

Abstract

We consider continuous state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term extinction and explosion behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and, as in the case of branching processes in random environment, we find five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, we study the process conditioned on eventual extinction where three regimes for the asymptotic behaviour of the extinction probability appear. Finally, the process conditioned on non-extinction and the process with immigration are given.