Small deviations of general Lévy processes

Abstract

We study the small deviation problem $\log \mathbb{P}({\sup }_{t\in [0,1]}|X_t|\le \epsilon )$, as $\epsilon \to 0$, for general Lévy processes $X$. The techniques enable us to determine the asymptotic rate for general real-valued Lévy processes, which we demonstrate with many examples. As a particular consequence, we show that a Lévy process with nonvanishing Gaussian component has the same (strong) asymptotic small deviation rate as the corresponding Brownian motion.

Published in at http://dx.doi.org/10.1214/09-AOP457 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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