Asymptotic Results for Heavy-tailed L\'evy Processes and their Exponential Functionals
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.
Cite
@article{arxiv.1912.04795,
title = {Asymptotic Results for Heavy-tailed L\'evy Processes and their Exponential Functionals},
author = {Wei Xu},
journal= {arXiv preprint arXiv:1912.04795},
year = {2020}
}
Comments
36 pages. Typos have been revised in this new version. Several proofs have been optimized. Several gaps in the proofs have been filled