 
Summary: The Annals of Statistics
2010, Vol. 38, No. 5, 30933128
DOI: 10.1214/09AOS749
© Institute of Mathematical Statistics, 2010
IS BROWNIAN MOTION NECESSARY TO MODEL
HIGHFREQUENCY DATA?
BY YACINE AÏTSAHALIA1 AND JEAN JACOD
Princeton University and UPMC (Université Paris6)
This paper considers the problem of testing for the presence of a con
tinuous 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 highfrequency individual
stock data, both tests point toward the need to include a continuous compo
nent in the model.
1. Introduction. This paper continues our development of statistical methods
designed to assess the specification of continuoustime models sampled at high fre
quency. The basic framework, inherited from theoretical models in mathematical
finance but also common in other fields such as physics or biology, is one where
the variable of interest X, in financial examples often the log of an asset price,
