Asymptotic results for exponential functionals of Levy processes

TL;DR

This paper provides a comprehensive analysis of the asymptotic behaviors of exponential functionals of Lévy processes, classifying their convergence rates and calculating precise limit constants, with applications to survival probabilities in branching processes.

Contribution

It offers a complete classification of asymptotic behaviors of exponential functionals of Lévy processes, including exact convergence speeds and limit constants, extending to applications in branching processes.

Findings

Five types of asymptotic behaviors identified

Exact convergence speeds determined

Limit constants explicitly calculated

Abstract

In this work we give a complete description to the asymptotic behaviors of exponential functionals of L\'evy processes and divide them into five different types according to their convergence rates. Not only their exact convergence speeds are proved, the accurate limit constants are also given. As an application, we study the survival probabilities of continuous-state branching processes in random environment defined in He et al. (2016). Like the discrete case and branching diffusion in random environment, we classify them into five different types according to their extinction speeds.