The Annals of Statistics

Testing for jumps in a discretely observed process

Yacine Aït-Sahalia and Jean Jacod

Full-text: Open access


We propose a new test to determine whether jumps are present in asset returns or other discretely sampled processes. As the sampling interval tends to 0, our test statistic converges to 1 if there are jumps, and to another deterministic and known value (such as 2) if there are no jumps. The test is valid for all Itô semimartingales, depends neither on the law of the process nor on the coefficients of the equation which it solves, does not require a preliminary estimation of these coefficients, and when there are jumps the test is applicable whether jumps have finite or infinite-activity and for an arbitrary Blumenthal–Getoor index. We finally implement the test on simulations and asset returns data.

Article information

Ann. Statist., Volume 37, Number 1 (2009), 184-222.

First available in Project Euclid: 16 January 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Jumps test discrete sampling high-frequency


Aït-Sahalia, Yacine; Jacod, Jean. Testing for jumps in a discretely observed process. Ann. Statist. 37 (2009), no. 1, 184--222. doi:10.1214/07-AOS568.

Export citation


  • Aït-Sahalia, Y. (2002). Telling from discrete data whether the underlying continuous-time model is a diffusion. J. Finance 57 2075–2112.
  • Aït-Sahalia, Y. (2004). Disentangling diffusion from jumps. J. Financial Economics 74 487–528.
  • Aït-Sahalia, Y. and Jacod, J. (2007). Volatility estimators for discretely sampled Lévy processes. Ann. Statist. 35 355–392.
  • Aït-Sahalia, Y. and Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. J. Financial Economics 83 413–452.
  • Andersen, T. G., Bollerslev, T. and Diebold, F. X. (2003). Some like it smooth, and some like it rough. Technical Report, Northwestern Univ.
  • Barndorff-Nielsen, O., Graversen, S., Jacod, J., Podolskij, M. and Shephard, N. (2006a). A central limit theorem for realised bipower variations of continuous semimartingales. In From Stochastic Calculus to Mathematical Finance, The Shiryaev Festschrift (Y. Kabanov, R. Liptser and J. Stoyanov, eds.) 33–69. Springer, Berlin.
  • Barndorff-Nielsen, O. E. and Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. J. Financial Econometrics 4 1–30.
  • Barndorff-Nielsen, O. E., Shephard, N. and Winkel, M. (2006b). Limit theorems for multipower variation in the presence of jumps. Stochastic Process. Appl. 116 796–806.
  • Carr, P. and Wu, L. (2003). What type of process underlies options? A simple robust test. J. Finance 58 2581–2610.
  • Huang, X. and Tauchen, G. (2006). The relative contribution of jumps to total price variance. J. Financial Econometrics 4 456–499.
  • Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559.
  • Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
  • Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Springer, New York.
  • Jiang, G. J. and Oomen, R. C. (2005). A new test for jumps in asset prices. Technical report, Univ. Warwick, Warwick Business School.
  • Lee, S. and Mykland, P. A. (2008). Jumps in financial markets: A new nonparametric test and jump clustering. Review of Financial Studies. To appear.
  • Lepingle, D. (1976). La variation d’ordre p des semi-martingales. Z. Wahrsch. Verw. Gebiete 36 295–316.
  • Mancini, C. (2001). Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell’Istituto Italiano degli Attuari LXIV 19–47.
  • Mancini, C. (2004). Estimating the integrated volatility in stochastic volatility models with Lévy type jumps. Technical report, Univ. Firenze.
  • Rényi, A. (1963). On stable sequences of events. Sankyā Ser. A 25 293–302.
  • Woerner, J. H. (2006a). Analyzing the fine structure of continuous-time stochastic processes. Technical Report, Univ. Göttingen.
  • Woerner, J. H. (2006b). Power and multipower variation: Inference for high-frequency data. In Stochastic Finance (A. Shiryaev, M. do Rosário Grosshino, P. Oliviera and M. Esquivel, eds.) 264–276. Springer, Berlin.