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.

Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

“…Moreover, and similarly to the case of BPREs, a phase transition in the subcritical regime appears. Recently, branching processes in a general Lévy random environment were introduced by Palau and Pardo in [28].…”

“…According to Palau and Pardo [28], we can define a self-similar branching process whose dynamics are affected by a Lévy random environment (SSBLRE) as the unique non-negative strong solution of the stochastic differential equation…”

“…The Laplace transform of Z t e −Kt can be computed explicitly and provides a closed formula for the probabilities of survival and non explosion of Z. Indeed according to Proposition 1 in [28], if (Z t , t ≥ 0) is a stable SSBLRE with index β ∈ (−1, 0) ∪ (0, 1], then for all z, λ > 0 and t ≥ 0,…”

“…Population model with competition in a Lévy random environment. We now study an extension of the competition model introduced in Evans et al [17] and studied by Palau and Pardo [28]. Following [28], we define a logistic process with competition in a Lévy random environment, (Z t , t ≥ 0), as the unique strong solution of the SDE…”

Asymptotic results for exponential functionals of Lévy processes

“…Here the random environment is modeled by the Lévy process {L(t) : t ≥ 0}. The reader may refer to He et al (2016) and Palau and Pardo (2015b) for discussions of more general CBRE-processes. We shall see that there is another Lévy process {ξ(t) : t ≥ 0} determined by the environment so that the survival probability of the CBRE-process up to time t ≥ 0 is given by P(X(t) > 0) = P 1 − exp − x(cα) −1/α A α t (ξ) −1/α .…”