Uniform Asymptotics for Compound Poisson Processes with Regularly Varying Jumps and Vanishing Drift

TL;DR

This paper develops uniform asymptotic estimates for the tail probabilities of functionals of spectrally one-sided Lévy processes with heavy tails, regular variation, and vanishing drift, extending results to near-critical regimes.

Contribution

It provides new uniform asymptotic results for various process functionals in near-critical regimes, using properties of scale functions.

Findings

Heavy-tailed large deviation estimates remain valid near criticality.

Asymptotics for supremum and exit times are established.

Results extend previous work on all-time supremum of such processes.

Abstract

This paper addresses heavy-tailed large deviation estimates for the distribution tail of functionals of a class of spectrally one-sided L\'evy process. Our contribution is to show that these estimates remain valid in a near-critical regime. This complements recent similar results that have been obtained for the all-time supremum of such processes. Specifically, we consider local asymptotics of the all-time supremum, the supremum of the process until exiting $[0,\infty)$, the maximum jump until that time, and the time it takes until exiting $[0,\infty)$. The proofs rely, among other things, on properties of scale functions.