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Seminar on Probability and Statistics |
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Seminar on Probability and Statistics Wednesday February 6 2008 Tokyo 056 2:50-4:00 pm
Estimating the Degree of Activity of jumps in High Frequency Data
Jean JACOD Universite Paris 6 Abstract Suppose that a continuous-time process X = (X_t )_{t >= 0}
is observed at finitely
many times, regularly spaced, on the fixed time interval [0, T ]. We suppose that
this process is an It\^o semimartingale, with a non-vanishing diffusion coefficient,
and with jumps. The aim is to estimate the so-called ”Blumenthal-Getoor”
index of the (partially observed) path on [0, T ], which is the (random) infimum
of all reals r such that the sum
\sum_{s\le T} |\Delta X_s|^r is finite (\Delta X_s denotes the jump
size at time s). When X is a L'evy process, this infimum is non-random, and
also independent of T , and has been introduced by Blumenthal and Getoor.
Under appropriate assumptions, unfortunately rather restrictive, we provide
an estimator, which is consistent when the step size between observations goes
to 0, and satisfies in addition a Central Limit Theorem. We also show the
(surprising) values that this estimator takes, when applied to real financial data.
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