Asymptotic behaviour of exponential functionals of L\'evy processes with applications to random processes in random environment

TL;DR

This paper investigates the long-term behavior of exponential functionals of Lévy processes, revealing five regimes based on the Laplace exponent, with applications to stochastic processes in random environments such as branching processes and diffusion maxima.

Contribution

It introduces a comprehensive analysis of the asymptotic behavior of exponential functionals of Lévy processes, including five regimes, and applies these results to various stochastic models in random environments.

Findings

Identified five regimes depending on the Laplace exponent shape.

Derived asymptotic behaviors for exponential functionals in different regimes.

Applied results to branching processes, population models, and diffusion maxima.

Abstract

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 important applications to stochastic processes in random environment, we study the asymptotic behaviour of \[ \mathbb{E}\Big[F\big(I_t(\xi)\big)\Big] \qquad \textrm{as}\qquad t\to \infty, \] where $F$ is a non-increasing function with polynomial decay at infinity and under some exponential moment conditions on $\xi$. In particular, we find five different regimes that depend on the shape of the Laplace exponent of $\xi$. Our proof relies on a discretisation of the exponential functional $I_t(\xi)$ and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply our main result to three {questions} associated to stochastic processes in random environment. We first consider the asymptotic behaviour of extinction and explosion for {stable} continuous state branching processes in a L\'evy random environment. Secondly, we {focus on} the asymptotic behaviour of the mean of a population model with competition in a L\'evy random environment and finally, we study the tail behaviour of the maximum of a diffusion process in a L\'evy random environment.