Estimation of the instantaneous volatility

This paper is concerned with the estimation of the volatility process in a stochastic volatility model of the following form: $d{X}_t=a_{t}dt+{\sigma }_{t}d{W}_t$, where $X$ denotes the log-price and $\sigma$ is a c`adl`ag semi-martingale. In the spirit of a series of recent works on the estimation of the cumulated volatility, we here focus on the instantaneous volatility for which we study estimators built as finite differences of the $\mathit{\text{power}}\mathit{\text{variations}}$ of the log-price. We provide central limit theorems with an optimal rate depending on the local behavior of $\sigma$. In particular, these theorems yield some confidence intervals for ${\sigma }_t$.