Volume 11, February 2007Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday
|Page(s)||327 - 343|
|Published online||17 August 2007|
Small ball probabilities for stable convolutions
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany; firstname.lastname@example.org
2 Equipe d'analyse et probabilités, Université d'Evry-Val d'Essonne, boulevard François Mitterrand, 91025 Evry Cedex, France; email@example.com
Revised: 10 July 2006
We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function with a real SαS Lévy process. We show that the small ball exponent is uniquely determined by the norm and by the behaviour of f at zero, which extends the results of Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 725–752 where this was proved for f being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 725–752 is obtained with respect to the norms, thanks to a weak decorrelation inequality due to Li, Elec. Comm. Probab. 4 (1999) 111–118. In the more difficult non-Gaussian case, we use a different method relying on comparison of entropy numbers and restrict ourselves to Hölder and Lp-norms.
Mathematics Subject Classification: 60F99 / 60G15 / 60G20 / 60G52
Key words: Entropy numbers / fractional Ornstein-Uhlenbeck processes / Riemann-Liouville processes / small ball probabilities / stochastic convolutions / wavelets.
© EDP Sciences, SMAI, 2007
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