QUARTICITY AND OTHER FUNCTIONALS OF VOLATILITY: EFFICIENT ESTIMATION

We consider a multidimensional Itô semimartingale regularly sampled on [0, t] at high frequency 1/. n , with . 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 ${\mathrm{\Delta }}_{\mathrm{n}}^{1/4}$ , this procedure reaches the parametric rate ${\mathrm{\Delta }}_{\mathrm{n}}^{1/2}$ , 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.