A measure-valued process describes the evolution of a population that evolves according to
the law of chance. In this chapter we provide some basic characterizations and constructions …

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 …

Let (Z n) be a supercritical branching process with immigration in a random environment.
Firstly, we prove that under a simple log moment condition on the offspring and immigration …

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 …

The asymptotic behavior of expectations of some exponential functionals of a Lévy process
is studied. The key point is the observation that the asymptotics only depend on the sample …

For a Lévy process ξ=(ξt) t⩾ 0 drifting to−∞, we define the so‐called exponential functional
as follows: I ξ=∫ 0∞ e ξ tdt. Under mild conditions on ξ, we show that the following …

Let $\xi=(\xi_t, t\ge 0) $ be a real-valued L\'evy process and define its associated exponential
functional as follows\[I_t (\xi):=\int_0^ t\exp\{-\xi_s\}{\rm d} s,\qquad t\ge 0.\] Motivated by …

We study the speed of extinction of continuous state branching processes in a Lévy
environment, where the associated Lévy process oscillates. Assuming that the Lévy process …

In this paper, we first provide several conditional limit theorems for Lévy processes with
negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior …

We continue with the systematic study of the speed of extinction of continuous-state
branching processes in Lévy environments under more general branching mechanisms …