## The Annals of Statistics

### Quarticity and other functionals of volatility: Efficient estimation

#### Abstract

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

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

Dates
First available in Project Euclid: 1 August 2013

https://projecteuclid.org/euclid.aos/1375362556

Digital Object Identifier
doi:10.1214/13-AOS1115

Mathematical Reviews number (MathSciNet)
MR3113818

Zentralblatt MATH identifier
1292.60033

#### Citation

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. https://projecteuclid.org/euclid.aos/1375362556

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