Cram\'er's estimate for stable processes with power drift

TL;DR

This paper analyzes the tail probabilities of the maximum of stable Lévy processes with power drift, revealing exponential decay in spectrally negative cases and polynomial decay otherwise, with explicit constants.

Contribution

It provides explicit asymptotic estimates for tail probabilities of stable Lévy processes with power drift, extending classical results to new process variants.

Findings

Exponential tail decay for spectrally negative stable processes.

Polynomial tail decay for other stable processes.

Explicit constants and exponents in tail asymptotics.

Abstract

We investigate the upper tail probabilities of the all-time maximum of a stable L\'evy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable L\'evy process. We also study the lower tail probabilities of the integrated stable L\'evy process in the presence of a power positive drift.