Annals of Statistics

Identifying the successive Blumenthal–Getoor indices of a discretely observed process

Yacine Aït-Sahalia and Jean Jacod

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Abstract

This paper studies the identification of the Lévy jump measure of a discretely-sampled semimartingale. We define successive Blumenthal–Getoor indices of jump activity, and show that the leading index can always be identified, but that higher order indices are only identifiable if they are sufficiently close to the previous one, even if the path is fully observed. This result establishes a clear boundary on which aspects of the jump measure can be identified on the basis of discrete observations, and which cannot. We then propose an estimation procedure for the identifiable indices and compare the rates of convergence of these estimators with the optimal rates in a special parametric case, which we can compute explicitly.

Article information

Source
Ann. Statist., Volume 40, Number 3 (2012), 1430-1464.

Dates
First available in Project Euclid: 5 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1346850061

Digital Object Identifier
doi:10.1214/12-AOS976

Mathematical Reviews number (MathSciNet)
MR3015031

Zentralblatt MATH identifier
1297.62051

Subjects
Primary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Keywords
Semimartingale Brownian motion jumps finite activity infinite activity discrete sampling high frequency

Citation

Aït-Sahalia, Yacine; Jacod, Jean. Identifying the successive Blumenthal–Getoor indices of a discretely observed process. Ann. Statist. 40 (2012), no. 3, 1430--1464. doi:10.1214/12-AOS976. https://projecteuclid.org/euclid.aos/1346850061


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References

  • Aït-Sahalia, Y. and Jacod, J. (2008). Fisher’s information for discretely sampled Lévy processes. Econometrica 76 727–761.
  • Aït-Sahalia, Y. and Jacod, J. (2009a). Estimating the degree of activity of jumps in high frequency financial data. Ann. Statist. 37 2202–2244.
  • Aït-Sahalia, Y. and Jacod, J. (2009b). Testing for jumps in a discretely observed process. Ann. Statist. 37 184–222.
  • Aït-Sahalia, Y. and Jacod, J. (2011). Testing whether jumps have finite or infinite activity. Ann. Statist. 39 1689–1719.
  • Aït-Sahalia, Y. and Jacod, J. (2012). Supplement to “Identifying the successive Blumenthal–Getoor indices of a discretely observed process.” DOI:10.1214/12-AOS976SUPP.
  • Basawa, I. V. and Brockwell, P. J. (1982). Nonparametric estimation for nondecreasing Lévy processes. J. Roy. Statist. Soc. Ser. B 44 262–269.
  • Belomestny, D. (2010). Spectral estimation of the fractional order of a Lévy process. Ann. Statist. 38 317–351.
  • Blumenthal, R. M. and Getoor, R. K. (1961). Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10 493–516.
  • Comte, F. and Genon-Catalot, V. (2009). Nonparametric estimation for pure jump Lévy processes based on high frequency data. Stochastic Process. Appl. 119 4088–4123.
  • Cont, R. and Mancini, C. (2011). Nonparametric tests for pathwise properties of semimartingales. Bernoulli 17 781–813.
  • Figueroa-López, J. E. and Houdré, C. (2006). Risk bounds for the non-parametric estimation of Lévy processes. In High Dimensional Probability (E. Giné, V. Koltchinskii, W. Li and J. Zinn, eds.). Institute of Mathematical Statistics Lecture Notes—Monograph Series 51 96–116. IMS, Beachwood, OH.
  • Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • Mancini, C. (2004). Estimating the integrated volatility in stochastic volatility models with Lévy type jumps. Technical report, Univ. Firenze.
  • Neumann, M. H. and Reiss, M. (2009). Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 223–248.
  • Nishiyama, Y. (2008). Nonparametric estimation and testing time-homogeneity for processes with independent increments. Stochastic Process. Appl. 118 1043–1055.
  • Rosiński, J. (2007). Tempering stable processes. Stochastic Process. Appl. 117 677–707.
  • Todorov, V. and Tauchen, G. (2010). Activity signature functions for high-frequency data analysis. J. Econometrics 154 125–138.
  • Zolotarev, V. M. (1995). On representation of densities of stable laws by special functions. Theory Probab. Appl. 39 354–362.

Supplemental materials

  • Supplementary material: Supplement to “Identifying the successive Blumenthal–Getoor indices of a discretely observed process”. This supplement contains the proof of Theorem 4.