Small deviations of general L\'{e}vy processes

TL;DR

This paper investigates the small deviation probabilities of general Lévy processes as the deviation parameter approaches zero, providing asymptotic rates and demonstrating their relation to Brownian motion.

Contribution

It introduces techniques to determine the asymptotic small deviation rates for general Lévy processes, including those with Gaussian components, and applies these to various examples.

Findings

Lévy processes with Gaussian components share the same small deviation rate as Brownian motion.

The paper provides explicit asymptotic rates for a wide class of Lévy processes.

Examples demonstrate the applicability of the theoretical results.

Abstract

We study the small deviation problem $\log\mathbb{P}(\sup_{t\in[0,1]}|X_t|\leq\varepsilon)$, as $\varepsilon\to0$, for general L\'{e}vy processes $X$. The techniques enable us to determine the asymptotic rate for general real-valued L\'{e}vy processes, which we demonstrate with many examples. As a particular consequence, we show that a L\'{e}vy process with nonvanishing Gaussian component has the same (strong) asymptotic small deviation rate as the corresponding Brownian motion.