  We provide a new non-parametric Fourier procedure to estimate the trajectory
of the instantaneous covariance process (from discrete observations of a
multidimensional price process) in the presence of jumps extending the seminal
work Malliavin and Mancino~\cite{MM:02, MM:09}. Our approach relies on a
modification of (classical) jump-robust estimators of integrated realized
covariance to estimate the Fourier coefficients of the covariance trajectory.
Using Fourier-F\'ejer inversion we reconstruct the path of the instantaneous
covariance. We prove consistency and central limit theorem (CLT) and in
particular that the asymptotic estimator variance is smaller by a factor $ 2/3$
in comparison to classical local estimators.
  The procedure is robust enough to allow for an iteration and we can show
theoretically and empirically how to estimate the integrated realized
covariance of the instantaneous stochastic covariance process. We apply these
techniques to robust calibration problems for multivariate modeling in finance,
i.e., the selection of a pricing measure by using time series and derivatives'
price information simultaneously.
