Quarticity and other functionals of volatility: efficient estimation

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 essentially \Delta_n^{1/4}, this procedure reaches the parametric rate \Delta_n^{1/2}. However, the limiting process in the central limit theorem exhibits a bias. 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.