Annals of Statistics

Testing whether jumps have finite or infinite activity

Yacine Aït-Sahalia and Jean Jacod

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We propose statistical tests to discriminate between the finite and infinite activity of jumps in a semimartingale discretely observed at high frequency. The two statistics allow for a symmetric treatment of the problem: we can either take the null hypothesis to be finite activity, or infinite activity. When implemented on high-frequency stock returns, both tests point toward the presence of infinite-activity jumps in the data.

Article information

Ann. Statist., Volume 39, Number 3 (2011), 1689-1719.

First available in Project Euclid: 25 July 2011

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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]

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


Aït-Sahalia, Yacine; Jacod, Jean. Testing whether jumps have finite or infinite activity. Ann. Statist. 39 (2011), no. 3, 1689--1719. doi:10.1214/11-AOS873.

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Supplemental materials

  • Supplementary material: Supplement to “Testing whether jumps have finite or infinite activity”. This supplementary article contains a few additional technical details about the assumptions made in this paper, and the proofs of all results.