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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
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