We assume that vectors x1 and x2 contain discrete observations from a bivariate diffusion process $(x_1,x_2)$ following the Ito stochastic differential equation $$dx_i(t)= \sigma_i(t) \ dW_i(t) + b_i(t) \ dt, \quad i=1,2,$$ where $W_1$ and $W_2$ are two Brownian motions defined on the filtered probability space $(\Omega, (\mathcal{F}_t)_{t \in [0,T]}, P)$, with correlation $\rho$, while $\sigma_1, \sigma_2, b_1$ and $b_2$ are random processes, adapted to $\mathcal{F}_t$.

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The integrated covariance of the process $(x_1,x_2)$ on $[0,T]$ is defined as $$IC_{[0,T]}:=\rho \, \int_0^T \sigma_1(t) \sigma_2(t) \, dt$$ Let $i=1,2$. 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 on $[0,T]$ is given by $$T c_0(c_{n_1,n_2,N})={T\over {2N+1}} \sum_{|k|\leq N} c_{k}(dx^1_{n_1})c_{-k}(dx^2_{n_2}),$$ $$c_k(dx^i_{n_i})= {1\over {T}} \sum_{l=0}^{n_i-1} e^{-{\rm i}\frac{2\pi}{T}kt^i_l}\delta_{l}(x_i).$$