Asymptotic Results for Heavy-tailed L\'evy Processes and their Exponential Functionals

TL;DR

This paper establishes asymptotic behaviors of heavy-tailed Lévy processes and their exponential functionals, revealing the influence of early large jumps and providing applications to branching processes in heavy-tailed environments.

Contribution

It offers new conditional limit theorems for heavy-tailed Lévy processes and characterizes the asymptotics of their exponential functionals, including explicit limit coefficients.

Findings

Asymptotics depend on early large jumps in sample paths.

Polynomial decay rates and exact limit coefficients are derived.

Application to extinction speed in heavy-tailed Lévy environments.

Abstract

In this paper we first provide several conditional limit theorems for L\'evy processes with negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior of expectations of some exponential functionals of heavy-tailed L\'evy processes. As the key point, we observe that the asymptotics mainly depends on the sample paths with early arrival large jump. Both the polynomial decay rate and the exact expression of the limit coefficients are given. As an application, we give an exact description for the extinction speed of continuous-state branching processes in heavy-tailed L\'evy random environment with stable branching mechanism.