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" nonparametric 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 $\min(\sqrt{n},(n\log n)^{(2-r)/2})$, where $n$ is the number of observations.