Speed of extinction for continuous state branching processes in subcritical L\'evy environments: the strongly and intermediate regimes

TL;DR

This paper investigates the extinction speed of continuous state branching processes in subcritical Lévy environments, extending previous work and analyzing the process behavior under different environmental drifts using advanced Lévy process techniques.

Contribution

It extends existing results on branching processes in Lévy environments by analyzing the extinction speed in subcritical regimes and characterizing the conditioned process.

Findings

Characterization of extinction speed in subcritical Lévy environments

Extension of previous results to broader Lévy process classes

Analysis of the conditioned process (Q-process) in this setting

Abstract

In this paper, we study the speed of extinction of continuous state branching processes in subcritical L\'evy environments. More precisely, when the associated L\'evy process to the environment drifts to $-\infty$ and, under a suitable exponential martingale change of measure (Esscher transform), the environment either drifts to $-\infty$ or oscillates. We extend recent results of Palau et al. (2016) and Li and Xu (2018), where the branching term is associated to a spectrally positive stable L\'evy process and complement the recent article of Bansaye et al. (2021) where the critical case was studied. Our methodology combines a path analysis of the branching process together with its L\'evy environment, fluctuation theory for L\'evy processes and the asymptotic behaviour of exponential functionals of L\'evy processes. As an application of the aforementioned results, we characterise the process conditioned to survival also known as the $Q$-process.