Continuous-state branching processes in Levy random environments

TL;DR

This paper introduces a comprehensive framework for continuous-state branching processes in Levy random environments, characterizing their transition dynamics, extinction probabilities, and ergodic behavior using stochastic integral equations.

Contribution

It provides new characterizations of CBRE-processes in Levy environments, including transition semigroups, extinction criteria, and ergodicity conditions, advancing understanding of their long-term behavior.

Findings

Process hits zero with positive probability under Grey's condition

Characterization of extinction probability via a singular differential equation

Established strong Feller property and ergodicity conditions

Abstract

A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a L\'evy process with no jump less than $-1$. We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another L\'evy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey's condition. In that case, a characterization of the extinction probability is given using a random differential equation with singular terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcricital CBRE process with immigration.