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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Abstract

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

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

Dates
First available in Project Euclid: 20 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1403276909

Digital Object Identifier
doi:10.1214/13-AOS1179

Mathematical Reviews number (MathSciNet)
MR3224283

Zentralblatt MATH identifier
1305.62036

Subjects
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

Keywords
Semimartingale volatility jumps infinite activity discrete sampling high frequency

Citation

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. https://projecteuclid.org/euclid.aos/1403276909


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References

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