Asymptotic results for heavy-tailed Lévy processes and their exponential functionals

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 of expectations of some exponential functionals of heavy-tailed Lévy processes. As the key point, we observe that the asymptotic mainly depends on the sample paths with early arrival of 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 Lévy random environment with stable branching mechanism.