Electronic Journal of Statistics

A note on central limit theorems for quadratic variation in case of endogenous observation times

Mathias Vetter and Tobias Zwingmann

Full-text: Open access

Abstract

This paper is concerned with a central limit theorem for quadratic variation when observations come as exit times from a regular grid. We discuss the special case of a semimartingale with deterministic characteristics and finite activity jumps in detail and illustrate technical issues in more general situations.

Article information

Source
Electron. J. Statist., Volume 11, Number 1 (2017), 963-980.

Dates
Received: May 2016
First available in Project Euclid: 30 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1490860813

Digital Object Identifier
doi:10.1214/17-EJS1252

Mathematical Reviews number (MathSciNet)
MR3629416

Zentralblatt MATH identifier
1361.60019

Subjects
Primary: 60F05: Central limit and other weak theorems 60G51: Processes with independent increments; Lévy processes 62M09: Non-Markovian processes: estimation

Keywords
High-frequency observations irregular data quadratic variation realized variance stable convergence

Rights
Creative Commons Attribution 4.0 International License.

Citation

Vetter, Mathias; Zwingmann, Tobias. A note on central limit theorems for quadratic variation in case of endogenous observation times. Electron. J. Statist. 11 (2017), no. 1, 963--980. doi:10.1214/17-EJS1252. https://projecteuclid.org/euclid.ejs/1490860813


Export citation

References

  • Bibinger, M. and M. Vetter (2015). Estimating the quadratic covariation of an asynchronously observed semimartingale with jumps., Ann. Inst. Statist. Math. 67(4), 707–743.
  • Borodin, A. N. and P. Salminen (2002)., Handbook of Brownian motion—facts and formulae (Second ed.). Probability and its Applications. Birkhäuser Verlag, Basel.
  • Freedman, D. (1983)., Brownian motion and diffusion (Second ed.). Springer-Verlag, New York-Berlin.
  • Fukasawa, M. (2010a). Central limit theorem for the realized volatility based on tick time sampling., Finance Stoch. 14(2), 209–233.
  • Fukasawa, M. (2010b). Realized volatility with stochastic sampling., Stochastic Process. Appl. 120(6), 829–852.
  • Fukasawa, M. and M. Rosenbaum (2012). Central limit theorems for realized volatility under hitting times of an irregular grid., Stochastic Process. Appl. 122(12), 3901–3920.
  • Hayashi, T., J. Jacod, and N. Yoshida (2011). Irregular sampling and central limit theorems for power variations: the continuous case., Ann. Inst. Henri Poincaré Probab. Stat. 47(4), 1197–1218.
  • Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales., Stochastic Process. Appl. 118(4), 517–559.
  • Jacod, J. and P. Protter (1998). Asymptotic error distributions for the Euler method for stochastic differential equations., Ann. Probab. 26(1), 267–307.
  • Jacod, J. and A. N. Shiryaev (2003)., Limit theorems for stochastic processes (Second ed.), Volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin.
  • Koike, Y. (2014). An estimator for the cumulative co-volatility of asynchronously observed semimartingales with jumps., Scand. J. Stat. 41(2), 460–481.
  • Li, Y., P. A. Mykland, E. Renault, L. Zhang, and X. Zheng (2014). Realized volatility when sampling times are possibly endogenous., Econometric Theory 30(3), 580–605.
  • Mitov, K. V. and E. Omey (2014)., Renewal processes. Springer Briefs in Statistics. Springer, Cham.
  • Mörters, P. and Y. Peres (2010)., Brownian motion. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge. With an appendix by Oded Schramm and Wendelin Werner.
  • Mykland, P. A. and L. Zhang (2012). The econometrics of high-frequency data. In, Statistical methods for stochastic differential equations, Volume 124 of Monogr. Statist. Appl. Probab., pp. 109–190. CRC Press, Boca Raton, FL.
  • Robert, C. Y. and M. Rosenbaum (2012). Volatility and covariation estimation when microstructure noise and trading times are endogenous., Math. Finance 22(1), 133–164.
  • Rosenbaum, M. and P. Tankov (2011). Asymptotic results for time-changed Lévy processes sampled at hitting times., Stochastic Process. Appl. 121(7), 1607–1632.