The Annals of Statistics

Estimating the degree of activity of jumps in high frequency data

Yacine Aït-Sahalia and Jean Jacod

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We define a generalized index of jump activity, propose estimators of that index for a discretely sampled process and derive the estimators’ properties. These estimators are applicable despite the presence of Brownian volatility in the process, which makes it more challenging to infer the characteristics of the small, infinite activity jumps. When the method is applied to high frequency stock returns, we find evidence of infinitely active jumps in the data and estimate their index of activity.

Article information

Ann. Statist., Volume 37, Number 5A (2009), 2202-2244.

First available in Project Euclid: 15 July 2009

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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]

Jumps index of activity infinite activity discrete sampling high frequency


Aït-Sahalia, Yacine; Jacod, Jean. Estimating the degree of activity of jumps in high frequency data. Ann. Statist. 37 (2009), no. 5A, 2202--2244. doi:10.1214/08-AOS640.

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