Weak limit theorems in the Fourier transform method for the estimation of multivariate volatility

In this paper, we prove some weak limit theorems for the Fourier estimator of multivariate volatility proposed by Malliavin and Mancino ( [12], [13]). We first give a central limit theorem for the estimator of the integrated volatility assuming that we observe the whole path of the Ito process. Then we study the case of discrete time observations possibly non synchronous. In this framework we prove that the asymptotic variance of the estimator depends on the limit behavior of the ratio N/n where N is the number of Fourier coefficients and n the number of observations. We point out some optimal choices of N with respect to n to minimize this asymptotic variance. MSC 2010. Primary: 62G20, Secondary: 60F05, 60H05.