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arXiv · 0805.1330

Small deviations of general Lévy processes

Frank Aurzada·Steffen Dereich
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Abstract

We study the small deviation problem $\log\mathbb{P}(\sup_{t\in[0,1]}|X_t|\leq\varepsilon)$, as $\varepsilon\to0$, 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. 2009-09-25. Small deviations of general Lévy processes. https://doi.org/10.1214/09-aop457

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