A remark on the rates of convergence for integrated volatility estimation in the presence of jumps

The optimal rate of convergence of estimators of the integrated volatility, for a discontinuous It\^o semimartingale sampled at regularly spaced times and over a fixed time interval, has been a long-standing problem, at least when the jumps are not summable. In this paper we study this optimal rate, in the minimax sense and for appropriate "bounded" non-parametric classes of semimartingales. We show that, if the $r$th powers of the jumps are summable for some $r\in[0,2)$, the minimax rate is equal to $\min(\rn,(n\log n)^{(2-r)/2})$, where $n$ is the number of observations.