Abstract
This chapter is concerned with nonparametric estimation of the Lévy density of a Lévy process. The sample path is observed at n equispaced instants with sampling interval Δ. We develop several nonparametric adaptive methods of estimation based on deconvolution, projection and kernel. The asymptotic framework is: n tends to infinity, Δ = Δ n tends to 0 while n Δ n tends to infinity (high frequency). Bounds for the \(\mathbb{L}^{2}\)-risk of estimators are given. Rates of convergence are discussed. Estimation of the drift and Gaussian component coefficients is studied. A specific method for estimating the jump density of compound Poisson processes is presented. Examples and simulation results illustrate the performance of estimators.
AMS Subject Classification 2000:
Primary: 62G05, 62M05
Secondary: 60G51
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aït-Sahalia, Y., & Jacod, J. (2007). Volatility estimators for discretely sampled Lévy processes. The Annals of Statistics, 35(1), 355–392.
Barndorff-Nielsen, O. E., & Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy processes (pp. 283–318). Boston, MA: Birkhäuser Boston.
Barndorff-Nielsen, O. E., Shephard, N., & Winkel, M. (2006). Limit theorems for multipower variation in the presence of jumps. Stochastic Processes and their Applications, 116(5), 796–806.
Barron, A., Birgé, L., & Massart, P. (1999). Risk bounds for model selection via penalization. Probability Theory and Related Fields, 113(3), 301–413.
Bec, M., & Lacour, C. (2014). Adaptive kernel estimation of the Lévy density. Statistical Inference for Stochastic Processes (to appear).
Belomestny, D., & Reiß, M. (2006). Spectral calibration of exponential Lévy models. Finance and Stochastics, 10(4), 449–474.
Bertoin, J. (1996). Lévy processes. Cambridge tracts in mathematics (Vol. 121). Cambridge: Cambridge University Press. ISBN: 0-521-56243-0.
Birgé, L., & Massart, P. (1998). Minimum contrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli, 4(3), 329–375.
Birgé, L., & Massart, P. (2007). Minimal penalties for Gaussian model selection. Probability Theory and Related Fields, 138(1–2), 33–73.
Buchmann, B., & Grübel, R. (2003). Decompounding: An estimation problem for Poisson random sums. The Annals of Statistics, 31(4), 1054–1074.
Chen, S. X., Delaigle, A., & Hall, P. (2010). Nonparametric estimation for a class of Lévy processes. The Journal of Econometrics, 157(2), 257–271.
Chesneau, C., Comte, F., & Navarro, F. (2013). Fast nonparametric estimation for convolutions of densities. Canadian Journal of Statistics, 41(4), 617–636.
Comte, F., Duval, C., & Genon-Catalot, V. (2014). Nonparametric density estimation in compound Poisson processes using convolution power estimators. Metrika, 77(1), 163–183.
Comte, F., & Genon-Catalot, V. (2009). Nonparametric estimation for pure jump Lévy processes based on high frequency data. Stochastic Processes and their Applications, 119(12), 4088–4123.
Comte, F., & Genon-Catalot, V. (2010). Non-parametric estimation for pure jump irregularly sampled or noisy Lévy processes. Statistica Neerlandica, 64(3), 290–313.
Comte, F., & Genon-Catalot, V. (2010). Nonparametric adaptive estimation for pure jump Lévy processes. Annales de Institut Henri Poincare (B) Probability and Statistics, 46(3), 595–617.
Comte, F., & Genon-Catalot, V. (2011). Estimation for Lévy processes from high frequency data within a long time interval. The Annals of Statistics, 39(2), 803–837.
Comte, F., & Lacour, C. (2011). Data-driven density estimation in the presence of additive noise with unknown distribution. Journal of the Royal Statistical Society Series B, 73(4), 601–627.
Comte, F., Rozenholc, Y., & Taupin, M.-L. (2006). Penalized contrast estimator for adaptive density deconvolution. Canadian Journal of Statistics, 34(3), 431–452.
Cont, R., & Tankov, P. (2004). Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Boca Raton, FL: Chapman & Hall/CRC. ISBN: 1-5848-8413-4.
DeVore, R. A., & Lorentz, G. G. (1993). Constructive approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences] (Vol. 303). Berlin: Springer. ISBN: 3-540-50627-6.
Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., & Picard, D. (1996). Density estimation by wavelet thresholding. The Annals of Statistics, 24(2), 508–539.
Duval, C. (2013). Density estimation for compound Poisson processes from discrete data. Stochastic Processes and their Applications, 123(11), 3963–3986.
Eberlein, E., & Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli, 1(3), 281–299.
Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events. Applications of mathematics (New York) (Vol. 33). Berlin: Springer. ISBN: 3-540-60931-8 (for insurance and finance).
Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. The Annals of Statistics, 19(3), 1257–1272.
Figueroa-López, J. E. (2008). Small-time moment asymptotics for Lévy processes. Statistics and Probability Letters, 78(18), 3355–3365.
Figueroa-López, J. E. (2009). Nonparametric estimation of Lévy models based on discrete-sampling. In Optimality. IMS lecture notes monograph series (Vol. 57, pp. 117–146). Beachwood, OH: Institute of Statistical Mathematics.
Figueroa-López, J. E., & Houdré, C. (2006). Risk bounds for the non-parametric estimation of Lévy processes. In High dimensional probability. IMS lecture notes monograph series (Vol. 51, pp. 96–116). Beachwood, OH: Institute of Statistical Mathematics.
Goldenshluger, A., & Lepski, O. (2011). Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. The Annals of Statistics, 39(3), 1608–1632.
Gugushvili, S. (2009). Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process. Journal of Nonparametric Statistics, 21(3), 321–343.
Gugushvili, S. (2012). Nonparametric inference for discretely sampled Lévy processes. Annales de l’Institut Henri Poincare (B) Probability and Statistics, 48(1), 282–307.
Hall, P., & Heyde, C. C. (1980). Martingale limit theory and its application. New York: Academic [Harcourt Brace Jovanovich Publishers]. ISBN: 0-12-319350-8. Probability and Mathematical Statistics.
Härdle, W., Kerkyacharian, G., Picard, D., & Tsybakov, A. (1998). Wavelets, approximation, and statistical applications. Lecture notes in statistics (Vol. 129). New York: Springer. ISBN: 0-387-98453-4.
Hirsch, F., & Lacombe, G. (1999). Elements of functional analysis. Graduate texts in mathematics (Vol. 192). New York: Springer. ISBN: 0-387-98524-7. Translated from the 1997 French original by Silvio Levy.
Jacod, J. (2007). Asymptotic properties of power variations of Lévy processes. ESAIM Probability and Statistics, 11, 173–196.
Johannes, J. (2009). Deconvolution with unknown error distribution. The Annals of Statistics, 37(5A), 2301–2323.
Jongbloed, G., van der Meulen, F. H., & van der Vaart, A. W. (2005). Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes. Bernoulli, 11(5), 759–791.
Kappus, J. (2014). Adaptive nonparametric estimation for Lévy processes observed at low frequency. Stochastic Process. Appl. 124(1), 730–758.
Kappus, J., & Reiß, M. (2010). Estimation of the characteristics of a Lévy process observed at arbitrary frequency. Statistica Neerlandica, 64(3), 314–328.
Katz, R. W. (2002). Stochastic modeling of hurricane damage. The Journal of Applied Meteorology, 41(7), 754–762.
Kerkyacharian, G., Lepski, O., & Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising. Probability Theory and Related Fields, 121(2), 137–170.
Klein, T., & Rio, E. (2005). Concentration around the mean for maxima of empirical processes. The Annals of Probability, 33(3), 1060–1077.
Küchler, U., & Tappe, S. (2008). Bilateral gamma distributions and processes in financial mathematics. Stochastic Processes and their Applications, 118(2), 261–283.
Lebedev, N. N. (1972). Special functions and their applications. New York: Dover Publications. Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication.
Madan, D. B., & Seneta, E. (1990). The variance gamma (v.g.) model for share market returns. The Journal of Business, 66(4), 511–524.
Massart, P. (2007). Concentration inequalities and model selection. Lecture notes in mathematics (Vol. 1896). Berlin: Springer. ISBN: 978-3-540-48497-4; 3-540-48497-3. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003, With a foreword by Jean Picard.
Meyer, Y. (1990). Ondelettes et opérateurs. I. Actualités Mathématiques [Current mathematical topics]. Paris: Hermann. ISBN: 2-7056-6125-0. Ondelettes [Wavelets].
Neumann, M. H. (1997). On the effect of estimating the error density in nonparametric deconvolution. Journal of Nonparametric Statistics, 7(4), 307–330.
Neumann, M. H., & Reiß, M. (2009). Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli, 15(1), 223–248.
Pensky, M., & Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. The Annals of Statistics, 27(6), 2033–2053.
Sato, K.-i. (1999). Lévy processes and infinitely divisible distributions. Cambridge studies in advanced mathematics (Vol. 68). Cambridge: Cambridge University Press. ISBN: 0-521-55302-4. Translated from the 1990 Japanese original, Revised by the author.
Scalas, E. (2006). The application of continuous-time random walks in finance and economics. Physica A, 362(2), 225–239.
Schick, A., & Wefelmeyer, W. (2004). Root n consistent density estimators for sums of independent random variables. Journal of Nonparametric Statistics, 16(6), 925–935.
Tsybakov, A. B. (2009). Introduction to nonparametric estimation. Springer series in statistics. New York: Springer. ISBN: 978-0-387-79051-0. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats.
Ueltzhöfer, F. A. J., & Klüppelberg, C. (2011). An oracle inequality for penalised projection estimation of Lévy densities from high-frequency observations. Journal of Nonparametric Statistics, 23(4), 967–989.
van Es, B., Gugushvili, S., & Spreij, P. (2007). A kernel type nonparametric density estimator for decompounding. Bernoulli, 13(3), 672–694.
Woerner, J. H. C. (2006). Power and multipower variation: Inference for high frequency data. In Stochastic finance (pp. 343–364). New York: Springer.
Zhang, L., Mykland, P. A., & Aït-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association, 100(472), 1394–1411.
Acknowledgements
We thank Céline Duval, as a coauthor of [13] which mainly corresponds to Sect. 10. We also wish to thank the referees for their careful reading and comments that helped improving the chapter. Last but not least, we are grateful to the editors of the Lévy Matters series for the invitation to contribute.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
The Talagrand Inequality The result below follows from the Talagrand concentration inequality given in [43] and arguments in [8] (see the proof of their Corollary 2 p. 354).
Lemma A.1 (Talagrand Inequality)
Let \(Y _{1},\ldots,Y _{n}\) be independent random variables, let \(\nu _{n,Y }(f) = (1/n)\sum _{i=1}^{n}[f(Y _{i}) - \mathbb{E}(f(Y _{i}))]\) and let \(\mathcal{F}\) be a countable class of uniformly bounded measurable functions. Then for ε2 > 0
with \(C(\epsilon ^{2}) = \sqrt{1 +\epsilon ^{2}} - 1\), K1 = 1∕6, and
By standard density arguments, this result can be extended to the case where \(\mathcal{F}\) is a unit ball of a linear normed space, after checking that \(f\mapsto \nu _{n}(f)\) is continuous and \(\mathcal{F}\) contains a countable dense family.
Lemma A.2 (The Rosenthal Inequality)
(see e.g. [33]) Let \((X_{i})_{1\leq i\leq n}\) be n independent centered random variables, such that \(\mathbb{E}(\vert X_{i}\vert ^{p}) < +\infty \) for an integer p ≥ 1. Then there exists a constant C(p) such that
Lemma A.3 (The Young Inequality)
(see [35]) Let f be a function belonging to \(\mathbb{L}^{p}(\mathbb{R})\) and g belonging to \(\mathbb{L}^{q}(\mathbb{R})\) , let p,q,r be real numbers in [1,+∞] and such that
Then
where f ⋆ g is the convolution product and \(\|f\|_{p}^{p} =\int \vert f(x)\vert ^{p}\mathit{dx}\) . In particular, for p = 1, r = q = 2, we have \(\|f \star g\|_{2} \leq \| f\|_{1}\;\|g\|_{2}\) .
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Comte, F., Genon-Catalot, V. (2015). Adaptive Estimation for Lévy Processes. In: Lévy Matters IV. Lecture Notes in Mathematics(), vol 2128. Springer, Cham. https://doi.org/10.1007/978-3-319-12373-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-12373-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12372-1
Online ISBN: 978-3-319-12373-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)