Quarticity and other functionals of volatility : Efficient estimation
Résumé
We consider a multidimensional Ito 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(1/4)(n), this procedure reaches the parametric rate Delta(1/2)(n), 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.