Non-asymptotic control of the cumulative distribution function of L\'evy processes

TL;DR

This paper develops non-asymptotic, optimal bounds for the tail probabilities of Lévy processes, applicable to a broad class with bounded Lévy densities near zero, enhancing understanding of their distributional behavior.

Contribution

It introduces non-asymptotic, optimal bounds for the tail distribution of Lévy processes with bounded densities, extending the analysis to a large class of such processes.

Findings

Provides explicit bounds for tail probabilities of Lévy processes.

Applicable to processes with Lévy densities bounded by stable-type densities.

Results are non-asymptotic and optimal.

Abstract

We propose non-asymptotic controls of the cumulative distribution function $P(|X_{t}|\ge \varepsilon)$, for any $t>0$, $\varepsilon>0$ and any L\'evy process $X$ such that its L\'evy density is bounded from above by the density of an $\alpha$-stable type L\'evy process in a neighborhood of the origin. The results presented are non-asymptotic and optimal, they apply to a large class of L\'evy processes.