The Annals of Statistics

Quarticity and other functionals of volatility: Efficient estimation

Jean Jacod and Mathieu Rosenbaum

Full-text: Open access


We consider a multidimensional Itô semimartingale regularly sampled on $[0,t]$ at high frequency $1/\Delta_{n}$, with $\Delta_{n}$ going to zero. The goal of this paper is to provide an estimator for the integral over $[0,t]$ of a given function of the volatility matrix. To approximate the integral, we simply use a Riemann sum based on local estimators of the pointwise volatility. We show that although the accuracy of the pointwise estimation is at most $\Delta_{n}^{1/4}$, this procedure reaches the parametric rate $\Delta_{n}^{1/2}$, as it is usually the case in integrated functionals estimation. After a suitable bias correction, we obtain an unbiased central limit theorem for our estimator and show that it is asymptotically efficient within some classes of sub models.

Article information

Ann. Statist., Volume 41, Number 3 (2013), 1462-1484.

First available in Project Euclid: 1 August 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G44: Martingales with continuous parameter 62F12: Asymptotic properties of estimators

Semimartingale high frequency data volatility estimation central limit theorem efficient estimation


Jacod, Jean; Rosenbaum, Mathieu. Quarticity and other functionals of volatility: Efficient estimation. Ann. Statist. 41 (2013), no. 3, 1462--1484. doi:10.1214/13-AOS1115.

Export citation


  • [1] Alvarez, A., Panloup, F., Pontier, M. and Savy, N. (2012). Estimation of the instantaneous volatility. Stat. Inference Stoch. Process. 15 27–59.
  • [2] Bickel, P. J. and Ritov, Y. (2003). Nonparametric estimators which can be “plugged-in”. Ann. Statist. 31 1033–1053.
  • [3] Birgé, L. and Massart, P. (1995). Estimation of integral functionals of a density. Ann. Statist. 23 11–29.
  • [4] Clément, E., Delattre, S. and Gloter, A. (2013). An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility. Stochastic Process. Appl. 123 2500–2521.
  • [5] Jacod, J. and Protter, P. (2012). Discretization of Processes. Stochastic Modelling and Applied Probability 67. Springer, Heidelberg.
  • [6] Jacod, J. and Reiß, M. (2013). A remark on the rates of convergence for integrated volatility estimation in the presence of jumps. Preprint. Available at arXiv:1209.4173.
  • [7] Jacod, J. and Rosenbaum, M. (2012). Estimation of volatility functionals: The case of a $\sqrt{n}$ window. Technical report, Laboratoire de Probabilités et Modèles Aléatoires, Univ. Pierre et Marie Curie. Available at arXiv:1212.1997.
  • [8] 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.
  • [9] Mancini, C. (2009). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scand. J. Stat. 36 270–296.
  • [10] Mancini, C. (2011). The speed of convergence of the threshold estimator of integrated variance. Stochastic Process. Appl. 121 845–855.
  • [11] Mykland, P. A. and Zhang, L. (2009). Inference for continuous semimartingales observed at high frequency. Econometrica 77 1403–1445.
  • [12] Vetter, M. (2010). Limit theorems for bipower variation of semimartingales. Stochastic Process. Appl. 120 22–38.