Weighted realized variance (${\hat{\Lambda }}_n(g)$)

For observations of an Itô integral on a partition of an interval, the statistic ${\sum }_{i}g(t_{i-1})(X_{t_i}-X_{t_{i-1}}{)}^2$ estimates $\int g(t)\sigma (t{)}^{2}dt$. With a deterministic noise coefficient the squared increments are independent, and the fluctuation has variance $2{\sum }_{i}g(t_{i-1}{)}^2({\int }_{t_{i-1}}^{t_i}{\sigma }^2{)}^2$.

The spot variance is the instantaneous squared noise coefficient $\sigma (t{)}^2$ of an Itô integral. It can be estimated by averaging the weighted realized variance over a shrinking time window. A Hölder continuity assumption bounds the smoothing bias.