We assume that the vectors x contains discrete observations from a diffusion process $x$ following the Ito stochastic differential equation $$dx(t)= \sigma(t) \ dW(t) + b(t) \ dt,$$ $$d\sigma^2(t)= \gamma(t) \ dZ(t) + a(t) \ dt,$$ where $W$ and $Z$ are two Brownian motions defined on the filtered probability space $(\Omega, (\mathcal{F}_t)_{t \in [0,T]}, P)$, with correlation $\rho$, while $\sigma, \gamma, b$ and $a$ are random processes, adapted to $\mathcal{F}_t$.

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The integrated volatility of volatility on $[0,T]$ is defined as $$ IVV_{[0,T]}:= \langle \sigma^2 ,\sigma^2 \rangle_T =\int_0^T\gamma^2(s)ds.$$ For any positive integer $n$, let $\mathcal{S}_{n}:=\{ 0=t_{0}\leq \cdots \leq t_{n}=T \}$ be the observation times. Moreover, let $\delta_l(x):= x(t_{l+1})-x(t_l)$ be the increments of $x$.

The Fourier estimator of the integrated volatility of volatility on $[0,T]$ is given by $$T c_0(V_{{n},N,M})={T^2\over {2M+1}} \sum_{|k|\leq M} c_{k}(d\sigma_{n,N})c_{-k}(d\sigma_{n,N}),$$ where: $$c_k(d\sigma_{n,N})= i k \frac{2\pi}{T} c_k(\sigma_{n,N}), \quad c_k(\sigma_{{n},N})={T\over {2N+1}} \sum_{|s|\leq N} c_{s}(dx_{n})c_{k-s}(dx_{n}),$$ $$c_k(dx_{n})= {1\over {T}} \sum_{l=0}^{n-1} e^{-{\rm i}\frac{2\pi}{T}kt_l}\delta_{l}(x).$$