Open Access
September 2009 Small deviations of general Lévy processes
Frank Aurzada, Steffen Dereich
Ann. Probab. 37(5): 2066-2092 (September 2009). DOI: 10.1214/09-AOP457


We study the small deviation problem logℙ(sup t∈[0, 1]|Xt|≤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.


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Frank Aurzada. Steffen Dereich. "Small deviations of general Lévy processes." Ann. Probab. 37 (5) 2066 - 2092, September 2009.


Published: September 2009
First available in Project Euclid: 21 September 2009

zbMATH: 1187.60035
MathSciNet: MR2561441
Digital Object Identifier: 10.1214/09-AOP457

Primary: 60G51

Keywords: Esscher transform , Lévy process , Lower tail probability , small ball problem , Small deviations

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 5 • September 2009
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