We assume our timeseries data are discrete observations from a diffusion process $x$ following the It\^o stochastic differential equation $$dx(t)= \sigma(t) \ dW(t) + b(t) \ dt,$$ where $W$ is a Brownian motion on a filtered probability space. Let $\sigma$ and $b$ be random processes, adapted to the Brownian filtration.

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The squared diffusion coefficient $\sigma^2(t)$ is called instantaneous variance of the process $x$.

For any positive integer $n$, let ${\cal S}_{n}:=\{ 0=t_{0}\leq \cdots \leq t_{n}=T \}$ be the observation times. Moreover, let $\delta_i(x):= x(t_{i+1})-x(t_i)$ be the increments of $x$.

The Fourier estimator of the instantaneous variance for $t \in (0,T)$, is defined as $$\widehat \sigma^2_{n,N,M}(t):= \sum_{|k|\leq M} \left(1- {|k|\over M}\right) e^{{\rm i} {{2\pi}\over {T}} tk} c_k(\sigma^2_{n,N}),$$ where $$c_k(\sigma^2_{n,N}):= {T \over {2N+1}}\sum_{|s|\leq N} c_s(dx_n) c_{k-s}(dx_n),$$ where for any integer $s$, $|s|\leq 2N$, the discretized Fourier coefficients of the increments are $$c_s(dx_{n}):= {1\over {T}} \sum_{i=0}^{n-1} e^{-{\rm i} {{2\pi}\over {T}} st_i}\delta_i(x).$$