Small deviations of general Lévy processes

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Small deviations of general Lévy processes

Frank Aurzada and Steffen Dereich

Source: Ann. Probab. Volume 37, Number 5 (2009), 2066-2092.

Abstract

We study the small deviation problem logℙ(sup _{t∈[0, 1]}|X_t|≤ɛ), as ɛ→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.

Primary Subjects: 60G51

Keywords: Small deviations; small ball problem; lower tail probability; Lévy process; Esscher transform

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Permanent link to this document: http://projecteuclid.org/euclid.aop/1253539864
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