The Annals of Statistics

Is Brownian motion necessary to model high-frequency data?

Yacine Aït-Sahalia and Jean Jacod

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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

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

First available in Project Euclid: 13 September 2010

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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. Is Brownian motion necessary to model high-frequency data?. Ann. Statist. 38 (2010), no. 5, 3093--3128. doi:10.1214/09-AOS749.

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