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: dX _t =  a _t dt + σ _t dW _t , where X denotes the log-price and σ is a càdlàg 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 power variations of the log-price. We provide central limit theorems with an optimal rate depending on the local behavior of σ. In particular, these theorems yield some confidence intervals for σ _t .