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June 2014 Efficient estimation of integrated volatility in presence of infinite variation jumps
Jean Jacod, Viktor Todorov
Ann. Statist. 42(3): 1029-1069 (June 2014). DOI: 10.1214/14-AOS1213

Abstract

We propose new nonparametric estimators of the integrated volatility of an Itô semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally “stable” Lévy processes, that is, processes whose Lévy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them.

Citation

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Jean Jacod. Viktor Todorov. "Efficient estimation of integrated volatility in presence of infinite variation jumps." Ann. Statist. 42 (3) 1029 - 1069, June 2014. https://doi.org/10.1214/14-AOS1213

Information

Published: June 2014
First available in Project Euclid: 20 May 2014

zbMATH: 1305.62146
MathSciNet: MR3210995
Digital Object Identifier: 10.1214/14-AOS1213

Subjects:
Primary: 60F05 , 60F17
Secondary: 60G07 , 60G51

Keywords: central limit theorem , integrated volatility , Itô semimartingale , Quadratic Variation

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 3 • June 2014
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