The Annals of Statistics

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

Jean Jacod and Markus Reiss

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The optimal rate of convergence of estimators of the integrated volatility, for a discontinuous Itô 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.

Article information

Ann. Statist., Volume 42, Number 3 (2014), 1131-1144.

First available in Project Euclid: 20 June 2014

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Zentralblatt MATH identifier

Primary: 62C20: Minimax procedures 62G20: Asymptotic properties 62M09: Non-Markovian processes: estimation
Secondary: 60H99: None of the above, but in this section 60J75: Jump processes

Semimartingale volatility jumps infinite activity discrete sampling high frequency


Jacod, Jean; Reiss, Markus. A remark on the rates of convergence for integrated volatility estimation in the presence of jumps. Ann. Statist. 42 (2014), no. 3, 1131--1144. doi:10.1214/13-AOS1179.

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