Small deviations of general Lévy processes

Details Small deviations of general Lévy processes Small deviations of general Lévy processes

2008-05-09

arxiv.org/abs/0805.1330v2

Annals of Probability 2009, Vol. 37, No. 5, 2066-2092

Mathematics

Probability

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

Scientific paper

10.1214/09-AOP457

We study the small deviation problem $\log\mathbb{P}(\sup_{t\in[0,1]}|X_t|\leq\varepsilon)$, as $\varepsilon\to0$, for general L\'{e}vy processes $X$. The techniques enable us to determine the asymptotic rate for general real-valued L\'{e}vy processes, which we demonstrate with many examples. As a particular consequence, we show that a L\'{e}vy process with nonvanishing Gaussian component has the same (strong) asymptotic small deviation rate as the corresponding Brownian motion.

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