 
Summary: The Annals of Statistics
2009, Vol. 37, No. 1, 184222
DOI: 10.1214/07AOS568
© Institute of Mathematical Statistics, 2009
TESTING FOR JUMPS IN A DISCRETELY OBSERVED PROCESS
BY YACINE AÏTSAHALIA1 AND JEAN JACOD
Princeton University and Université Pierre et Marie Curie
We propose a new test to determine whether jumps are present in asset
returns or other discretely sampled processes. As the sampling interval tends
to 0, our test statistic converges to 1 if there are jumps, and to another deter
ministic and known value (such as 2) if there are no jumps. The test is valid
for all Itô semimartingales, depends neither on the law of the process nor on
the coefficients of the equation which it solves, does not require a prelim
inary estimation of these coefficients, and when there are jumps the test is
applicable whether jumps have finite or infiniteactivity and for an arbitrary
BlumenthalGetoor index. We finally implement the test on simulations and
asset returns data.
1. Introduction. The problem of deciding whether the continuoustime
process which models an economic or financial time series should have contin
uous paths or exhibit jumps is becoming an increasingly important issue, in view
