Testing for jumps in a discretely observed process. (English) Zbl 1155.62057

Summary: 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 the jumps have finite or infinite-activity and for an arbitrary Blumenthal-Getoor index. We finally implement the test on simulations and asset returns data.


62M02 Markov processes: hypothesis testing
62F12 Asymptotic properties of parametric estimators
60G48 Generalizations of martingales
62M05 Markov processes: estimation; hidden Markov models
60F05 Central limit and other weak theorems
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI arXiv Euclid


[1] Aït-Sahalia, Y. (2002). Telling from discrete data whether the underlying continuous-time model is a diffusion. J. Finance 57 2075-2112.
[2] Aït-Sahalia, Y. (2004). Disentangling diffusion from jumps. J. Financial Economics 74 487-528.
[3] Aït-Sahalia, Y. and Jacod, J. (2007). Volatility estimators for discretely sampled Lévy processes. Ann. Statist. 35 355-392. · Zbl 1114.62109 · doi:10.1214/009053606000001190
[4] Aït-Sahalia, Y. and Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. J. Financial Economics 83 413-452.
[5] Andersen, T. G., Bollerslev, T. and Diebold, F. X. (2003). Some like it smooth, and some like it rough. Technical Report, Northwestern Univ. · Zbl 1142.91712
[6] 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. · Zbl 1106.60037
[7] 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.
[8] 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. · Zbl 1096.60022 · doi:10.1016/j.spa.2006.01.007
[9] Carr, P. and Wu, L. (2003). What type of process underlies options? A simple robust test. J. Finance 58 2581-2610.
[10] Huang, X. and Tauchen, G. (2006). The relative contribution of jumps to total price variance. J. Financial Econometrics 4 456-499.
[11] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517-559. · Zbl 1142.60022 · doi:10.1016/j.spa.2007.05.005
[12] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267-307. · Zbl 0937.60060 · doi:10.1214/aop/1022855419
[13] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes , 2nd ed. Springer, New York. · Zbl 1018.60002
[14] Jiang, G. J. and Oomen, R. C. (2005). A new test for jumps in asset prices. Technical report, Univ. Warwick, Warwick Business School.
[15] Lee, S. and Mykland, P. A. (2008). Jumps in financial markets: A new nonparametric test and jump clustering. Review of Financial Studies.
[16] Lepingle, D. (1976). La variation d’ordre p des semi-martingales. Z. Wahrsch. Verw. Gebiete 36 295-316. · Zbl 0325.60047 · doi:10.1007/BF00532696
[17] Mancini, C. (2001). Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell’Istituto Italiano degli Attuari LXIV 19-47.
[18] Mancini, C. (2004). Estimating the integrated volatility in stochastic volatility models with Lévy type jumps. Technical report, Univ. Firenze.
[19] Rényi, A. (1963). On stable sequences of events. Sankyā Ser. A 25 293-302. · Zbl 0141.16401
[20] Woerner, J. H. (2006a). Analyzing the fine structure of continuous-time stochastic processes. Technical Report, Univ. Göttingen. · Zbl 1416.62592
[21] 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. · Zbl 1142.62095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.