  We provide a nonparametric method for the computation of instantaneous
multivariate volatility for continuous semi-martingales, which is based on
Fourier analysis. The co-volatility is reconstructed as a stochastic function
of time by establishing a connection between the Fourier transform of the
prices process and the Fourier transform of the co-volatility process. A
nonparametric estimator is derived given a discrete unevenly spaced and
asynchronously sampled observations of the asset price processes. The
asymptotic properties of the random estimator are studied: namely, consistency
in probability uniformly in time and convergence in law to a mixture of
Gaussian distributions.
