Branching processes in a Lévy random environment

Abstract

In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by Brownian motions and Poisson random measures which are mutually independent. The existence and uniqueness of strong solutions are established under some general conditions that allows us to consider the case when the strong solution explodes at a finite time. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study some properties of such processes that extends recent results obtained by Bansaye et al. in (Electron. J. Probab. 18, no. 106, 1-31, (2013)), Palau and Pardo in (arXiv:1506.09197 (2015)) and Evans et al. in (J. Math. Biol., 71, 325-359, (2015)).

New proof of main theorem

Cited by in corpus (5)

• Extinction rate of continuous state branching processes in critical Lévy environments
• Scaling limits of general population processes - Wright-Fisher and branching processes in random environment
• Asymptotic results for exponential functionals of Levy processes
• Continuous-state branching processes in Levy random environments
• Asymptotic Results for Heavy-tailed Lévy Processes and their Exponential Functionals