We assume that vectors xi contain discrete observations from a d-dimensional diffusion process $(x_1,..,x_d)$ following the Ito stochastic differential equation $$dx_i(t)= \sigma_i(t) \ dW_i(t) + b_i(t) \ dt, \quad i=1,...,d,$$ where, for all i, $W_i$ is a Brownian motion defined on the filtered probability space $(\Omega, (\mathcal{F}_t)_{t \in [0,T]}, P)$, while $\sigma_i$ and $b_i$ are random processes, adapted to $\mathcal{F}_t$.

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The integrated covariance of the process $(x_i,x_j)$ on $[0,T]$ is defined as $$IC_{ij,[0,T]}:=\rho_{ij} \, \int_0^T \sigma_i(t) \sigma_j(t) \, dt$$ where $\rho_{ij}$ denotes the correlation between $W_i$ and $W_j$.

For any positive integer $n_i$, let $\mathcal{S}^i_{n_i}:=\{ 0=t^i_{0}\leq \cdots \leq t^i_{n_i}=T \}$ be the observation times for the $i$-th asset. Moreover, let $\delta_l(x^i):= x^i(t^i_{l+1})-x^i(t^i_l)$ be the increments of $x^i$.

The Fourier estimator of the integrated covariance with fejer kernel on $[0,T]$ is given by $$T c_0(c_{n_i,n_j,N})={T^2\over {N+1}} \sum_{|k|\leq N} \left( 1- \frac{|k|}{N+1}\right)c_{k}(dx^i_{n_1})c_{-k}(dx^j_{n_2}),$$ $$c_k(dx^i_{n_i})= {1\over {T}} \sum_{l=0}^{n_i-1} e^{-{\rm i}\frac{2\pi}{T}kt^i_l}\delta_{l}(x_i).$$