We assume our timeseries data are discrete observations from a diffusion
process following the Ito 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.
See the Reference for further mathematical details.
The integrated variance of the process over the time interval [0,T] is defined as
\int_0^T \sigma^2(t) dt.
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 Fejer-Fourier estimator of the integrated variance over [0,T], is
defined as
\widehat\sigma^{2}_{n,N}:= {T^2 \over {N+1}}\sum_{|s|\leq N} \left( 1-
{{|s|}\over {N}} \right)c_s(dx_n) c_{-s}(dx_n),
where for any integer k, |k|\leq N, the discretized Fourier
coefficients of the increments are
c_k(dx_{n}):= {1\over {T}} \sum_{i=0}^{n-1} e^{-{\rm i} {{2\pi}\over {T}}
kt_i}\delta_i(x).
The cutting frequency N is a scalar integer. If not specified, N is
set equal to n/2 (Nyquist frequency).