The Annals of Statistics

Is Brownian motion necessary to model high-frequency data?

Yacine Aït-Sahalia and Jean Jacod

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Abstract

This paper considers the problem of testing for the presence of a continuous part in a semimartingale sampled at high frequency. We provide two tests, one where the null hypothesis is that a continuous component is present, the other where the continuous component is absent, and the model is then driven by a pure jump process. When applied to high-frequency individual stock data, both tests point toward the need to include a continuous component in the model.

Article information

Source
Ann. Statist., Volume 38, Number 5 (2010), 3093-3128.

Dates
First available in Project Euclid: 13 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1284391759

Digital Object Identifier
doi:10.1214/09-AOS749

Mathematical Reviews number (MathSciNet)
MR2722465

Zentralblatt MATH identifier
1327.62118

Subjects
Primary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Keywords
Semimartingale Brownian motion jumps finite activity infinite activity discrete sampling high frequency

Citation

Aït-Sahalia, Yacine; Jacod, Jean. Is Brownian motion necessary to model high-frequency data?. Ann. Statist. 38 (2010), no. 5, 3093--3128. doi:10.1214/09-AOS749. https://projecteuclid.org/euclid.aos/1284391759


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References

  • [1] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover, New York.
  • [2] Aït-Sahalia, Y. and Jacod, J. (2008). Fisher’s information for discretely sampled Lévy processes. Econometrica 76 727–761.
  • [3] Aït-Sahalia, Y. and Jacod, J. (2008). Testing whether jumps have finite or infinite activity. Technical report, Princeton Univ. and Univ. de Paris-6.
  • [4] Aït-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency financial data. Ann. Statist. 37 2202–2244.
  • [5] Aït-Sahalia, Y. and Jacod, J. (2009). Testing for jumps in a discretely observed process. Ann. Statist. 37 184–222.
  • [6] Ball, C. A. and Torous, W. N. (1983). A simplified jump process for common stock returns. Journal of Financial and Quantitative Analysis 18 53–65.
  • [7] Bates, D. S. (1991). The crash of ’87: Was it expected? The evidence from options markets. Journal of Finance 46 1009–1044.
  • [8] Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. Journal of Business 75 305–332.
  • [9] Carr, P. and Wu, L. (2003). The finite moment log stable process and option pricing. Journal of Finance 58 753–777.
  • [10] Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1 281–299.
  • [11] Jacod, J. (2007). Statistics and high frequency data: SEMSTAT Seminar. Technical report, Univ. de Paris-6.
    Mathematical Reviews (MathSciNet): MR2320815
    Digital Object Identifier: doi:10.1051/ps:2007013
  • [12] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559.
    Mathematical Reviews (MathSciNet): MR2394762
    Zentralblatt MATH: 1142.60022
    Digital Object Identifier: doi:10.1016/j.spa.2007.05.005
  • [13] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1943877
  • [14] Madan, D. B. and Seneta, E. (1990). The Variance Gamma (V.G.) model for share market returns. Journal of Business 63 511–524.
  • [15] Mancini, C. (2001). Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell’Istituto Italiano degli Attuari LXIV 19–47.
  • [16] Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 125–144.
  • [17] Tauchen, G. T. and Todorov, V. (2009). Activity signature functions for high-frequency data analysis. Technical report, Duke Univ.
    Mathematical Reviews (MathSciNet): MR2558956
    Digital Object Identifier: doi:10.1016/j.jeconom.2009.06.009

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