Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-16T14:08:46.865Z Has data issue: false hasContentIssue false
  • English
  • Français

Small ball probabilities for stable convolutions

Published online by Cambridge University Press:  17 August 2007

Frank Aurzada  and
Thomas Simon
Frank Aurzada
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany; aurzada@math.tu-berlin.de
Thomas Simon
Affiliation:
Equipe d'analyse et probabilités, Université d'Evry-Val d'Essonne, boulevard François Mitterrand, 91025 Evry Cedex, France; tsimon@univ-evry.fr
Get access

Abstract

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function $f : \; ]0, +\infty[ \;\to \mathbb{R}$ 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.

Keywords

Entropy numbersfractional Ornstein-Uhlenbeck processesRiemann-Liouville processessmall ball probabilitiesstochastic convolutionswavelets.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 327 - 343
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Artstein, S., Milman, V. and Szarek, S.J., Duality of metric entropy. Ann. Math. (2) 159 (2004) 1213–1328.
F. Aurzada., Metric entropy and the small deviation problem for stable processes. To appear in Prob. Math. Stat. (2006).
Berthet, P. and Shi, Z., Small ball estimates for Brownian motion under a weighted sup-norm. Studia Sci. Math. Hungar. 36 (2000) 275–289.
Bertoin, J., On the first exit time of a completely asymmetric stable process from a finite interval. Bull. London Math. Soc. 28 (1996) 514–520. CrossRef
Cheridito, P., Kawaguchi, H. and Maejima, M., Fractional Ornstein-Uhlenbeck Processes. Elec. J. Probab. 8 (2003) 1–14. CrossRef
D.E. Edmunds and H. Triebel, Function spaces, entropy numbers, differential operators. Cambridge University Press (1996).
Hardy, G.H. and Littlewood, J.E., Some properties of fractional integrals I. Math. Z. 27 (1927) 565–606. CrossRef
Heath, D., Jarrow, R.A. and Morton, A., Bond pricing and the term structure of interest rate: A new methodology for contingent claim valuation. Econometrica 60 (1992) 77–105. CrossRef
Kuelbs, J. and Metric Entropy, W.V. Li and the Small Ball Problem for Gaussian Measures. J. Func. Anal. 116 (1993) 113–157. CrossRef
S. Kwapień, M.B. Marcus and J. Rosiński, Two results on continuity and boundedness of stochastic convolutions. Ann. Inst. Henri Poincaré, Probab. et Stat. 42 (2006) 553–566.
Li, W.V., Gaussian, A correlation inequality and its application to small ball probabilities. Elec. Comm. Probab. 4 (1999) 111–118. CrossRef
Small, W.V. Li ball probabilities for Gaussian Markov processes under the L p -norm. Stochastic Processes Appl. 92 (2001) 87–102.
Li, W.V. and Linde, W., Existence of small ball constants for fractional Brownian motions. C. R. Acad. Sci. Paris 326 (1998) 1329–1334. CrossRef
Li, W.V. and Linde, W., Approximation, metric entropy and the small ball problem for Gaussian measures. Ann. Probab. 27 (1999) 1556–1578.
Li, W.V. and Linde, W., Small Deviations of Stable Processes via Metric Entropy. J. Theoret. Probab. 17 (2004) 261–284. CrossRef
Lifshits, M.A. and Simon, T., Small deviations for fractional stable processes. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 725–752. CrossRef
Maejima, M. and Yamamoto, K., Long-Memory Stable Ornstein-Uhlenbeck Processes. Elec. J. Probab. 8 (2003) 1–18. CrossRef
S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach (1993).
G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes. Chapman & Hall (1994).
Takashima, K., Sample path properties of ergodic self-similar processes. Osaka J. Math. 26 (1989) 159–189.
Taqqu, M.S. and Wolpert, R.L., Fractional Ornstein-Uhlenbeck Lévy Processes and the Telecom Process: Upstairs and Downstairs. Signal Processing 85 (2005) 1523–1545.