## Bernoulli

• Bernoulli
• Volume 17, Number 2 (2011), 643-670.

### Sieve-based confidence intervals and bands for Lévy densities

José E. Figueroa-López

#### Abstract

The estimation of the Lévy density, the infinite-dimensional parameter controlling the jump dynamics of a Lévy process, is considered here under a discrete-sampling scheme. In this setting, the jumps are latent variables, the statistical properties of which can be assessed when the frequency and time horizon of observations increase to infinity at suitable rates. Nonparametric estimators for the Lévy density based on Grenander’s method of sieves was proposed in Figueroa-López [IMS Lecture Notes 57 (2009) 117–146]. In this paper, central limit theorems for these sieve estimators, both pointwise and uniform on an interval away from the origin, are obtained, leading to pointwise confidence intervals and bands for the Lévy density. In the pointwise case, our estimators converge to the Lévy density at a rate that is arbitrarily close to the rate of the minimax risk of estimation on smooth Lévy densities. In the case of uniform bands and discrete regular sampling, our results are consistent with the case of density estimation, achieving a rate of order arbitrarily close to $\log^{−1/2}(n) ⋅ n^{−1/3}$, where $n$ is the number of observations. The convergence rates are valid, provided that $s$ is smooth enough and that the time horizon $T_n$ and the dimension of the sieve are appropriately chosen in terms of $n$.

#### Article information

Source
Bernoulli, Volume 17, Number 2 (2011), 643-670.

Dates
First available in Project Euclid: 5 April 2011

https://projecteuclid.org/euclid.bj/1302009241

Digital Object Identifier
doi:10.3150/10-BEJ286

Mathematical Reviews number (MathSciNet)
MR2787609

Zentralblatt MATH identifier
1345.62061

#### Citation

Figueroa-López, José E. Sieve-based confidence intervals and bands for Lévy densities. Bernoulli 17 (2011), no. 2, 643--670. doi:10.3150/10-BEJ286. https://projecteuclid.org/euclid.bj/1302009241

#### References

• [1] Barron, A., Birgé, L. and Massart, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301–413.
• [2] Bertoin, J. (1996). Lévy Processes. Cambridge: Cambridge Univ. Press.
• [3] Bickel, P.J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095.
• [4] Birgé, L. and Massart, P. (1997). From model selection to adaptive estimation. In Festschrift for Lucien Le Cam 55–87. New York: Springer.
• [5] Brillinger, D.R. (1969). An asymptotic representation of the sample distribution function. Bull. Amer. Math. Soc. 75 545–547.
• [6] Carr, P., Geman, H., Madan, D. and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. J. Business 75 305–332.
• [7] Carr, P., Madan, D. and Chang, E. (1998). The variance Gamma process and option pricing. European Finance Rev. 2 79–105.
• [8] Chung, K.L. (2001). A Course in Probability Theory. San Diego, CA: Academic Press.
• [9] Cont, R. and Tankov, P. (2003). Financial Modelling with Jump Processes. Boca Raton, FL: Chapman & Hall.
• [10] Figueroa-López, J.E. (2004). Nonparametric estimation of Lévy processes with a view towards mathematical finance. Ph.D. thesis, Georgia Institute of Technology. Available at http://etd.gatech.edu, No. etd-04072004-122020.
• [11] Figueroa-López, J.E. (2009). Nonparametric estimation for Lévy models based on discrete-sampling. In Optimality: The Third Erich L. Lehmann Symposium 117–146. IMS Lecture Notes–Monograph Series 57. Beachwood, OH: IMS.
• [12] Figueroa-López, J.E. (2009). Nonparametric estimation of time-changed Lévy models under high-frequency data. Adv. Appl. Probab. 41 1161–1188.
• [13] Figueroa-López, J.E. (2010). Jump-diffusion models driven by Lévy processes. In Handbook of Computational Finance (J.-C. Duan, J.E. Gentle and W. Hardle, eds.). Springer. To appear.
• [14] Figueroa-López, J.E. and Houdré, C. (2006). Risk bounds for the non-parametric estimation of Lévy processes. In High Dimensional Probability 96–116. IMS Lecture Notes – Monograph Series 51. Beachwood, OH: IMS.
• [15] Figueroa-López, J.E. and Houdré, C. (2009). Small-time expansions for the transition distributions of Lévy processes. Stochastic Process. Appl. 119 3862–3889.
• [16] Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics. Melbourne, FL: Krieger.
• [17] Grenander, U. (1981). Abstract Inference. New York: Wiley.
• [18] Hall, P. (1992). Effect of bias estimation on coverage accuracy of bootstrap confidence interval for a probability density. Ann. Statist. 22 675–694.
• [19] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent RV’-s, and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
• [20] Madan, D.B. and Seneta, E. (1990). The variance Gamma model for share market returns. J. Business 63 511–524.
• [21] Prause, K. (1999). The generalized hyperbolic model: Estimation, financial derivatives, and risk measures. PhD thesis, Univ. Freiburg.
• [22] Rüschendorf, L. and Woerner, J. (2002). Expansion of transition distributions of Lévy processes in small time. Bernoulli 8 81–96.
• [23] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge Univ. Press.
• [24] Seneta, E. (2004). Fitting the variance-gamma model to financial data. J. Appl. Probab. 41A 177–187.
• [25] Woerner, J. (2003). Variational sums and power variation: A unifying approach to model selection and estimation in semimartingale models. Statist. Decisions 21 47–68.