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.
"Is Brownian motion necessary to model high-frequency data?." Ann. Statist. 38 (5) 3093 - 3128, October 2010. https://doi.org/10.1214/09-AOS749