The Annals of Probability

Small deviations of general Lévy processes

Frank Aurzada and Steffen Dereich

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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.

Article information

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

First available in Project Euclid: 21 September 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes

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


Aurzada, Frank; Dereich, Steffen. Small deviations of general Lévy processes. Ann. Probab. 37 (2009), no. 5, 2066--2092. doi:10.1214/09-AOP457.

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