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Summary: The Annals of Statistics
2009, Vol. 37, No. 5A, 22022244
DOI: 10.1214/08-AOS640
© Institute of Mathematical Statistics, 2009
ESTIMATING THE DEGREE OF ACTIVITY OF
JUMPS IN HIGH FREQUENCY DATA
BY YACINE AÏT-SAHALIA1 AND JEAN JACOD
Princeton University and UPMC (Université Paris-6)
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 fre-
quency stock returns, we find evidence of infinitely active jumps in the data
and estimate their index of activity.
1. Introduction. Using high frequency financial data, which are now widely
available, we can hope to answer a number of questions regarding the characteris-
tics of the process that drives asset returns. Let us model the log-price X of some
asset as a 1-dimensional process, which we will observe over a fixed time interval
[0,T ] at discrete times 0, n,2 n,... with a time interval n between succes-
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