## The Annals of Statistics

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

#### 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

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

Digital Object Identifier
doi:10.1214/13-AOS1179

Mathematical Reviews number (MathSciNet)
MR3224283

Zentralblatt MATH identifier
1305.62036

#### 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

#### References

• [1] Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J., Podolskij, M. and Shephard, N. (2006). A central limit theorem for realised power and bipower variations of continuous semimartingales. In From Stochastic Calculus to Mathematical Finance (Yu. Kabanov, R. Liptser and J. Stoyanov, eds.) 33–68. Springer, Berlin.
• [2] Barndorff-Nielsen, O. E. and Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps. J. Financ. Econom. 2 1–48.
• [3] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559.
• [4] Jacod, J. and Protter, P. (2012). Discretization of Processes. Springer, Heidelberg.
• [5] Mancini, C. (2001). Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. G. Ist. Ital. Attuari LXIV 19–47.
• [6] Mancini, C. (2011). The speed of convergence of the threshold estimator of integrated variance. Stochastic Process. Appl. 121 845–855.
• [7] Vetter, M. (2010). Limit theorems for bipower variation of semimartingales. Stochastic Process. Appl. 120 22–38.