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

See the References for further mathematical details.

The integrated quarticity on $[0,T]$ is defined as $$IQ_{[0,T]}:=\int_0^T\sigma^4(s)ds.$$ 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 quarticity on $[0,T]$ is given by $$T c_0(Q_{{n},N.M})={T^2\over {2M+1}} \sum_{|k|\leq M} c_{k}(\sigma_{n,N})c_{-k}(\sigma_{n,N}),$$ where: $$c_k(\sigma_{{n},N})={T\over {2N+1}} \sum_{|s|\leq N} c_{s}(dx_{n})c_{k-s}(dx_{n}),$$ $$c_k(dx_{n})= {1\over {T}} \sum_{l=0}^{n-1} e^{-{\rm i}\frac{2\pi}{T}kt_l}\delta_{l}(x).$$