Yacine Ait-Sahalia (Princeton University), Jean Jacod* (Universit¨¦ Paris)
For modelling purposes one would like to infer the characteristics (the drift, the - possibly stochastic - volatility and the - possibly stochastic - Levy measure) from discretely sampled observations from a semimartingale. When the sampling interval goes to 0, it is well known that one can infer consistently the volatility, under very weak assumptions. But such a consistent inference is impossible for the drift and also for the Levy measure, if the overall length of observation is kept fixed. In fact, even in the unrealistic case when the whole path of X is observed over [0,T], one can infer neither the drift nor the Levy measure. One can however decide about the behavior of the Levy measure near 0: first if it does not explode near 0, meaning that the number of jumps is finite; second, when this number is infinite, we can say something about the concentration of small jumps. For this purpose, we propose a generalization of the Blumenthal-Getoor index to semimartingales and construct consistent estimators of this index. These estimators are applicable despite the fact that the semimartingale has a continuous part, which makes it more challenging to learn about the small jumps. We can then test for instance the null hypotheses that jumps have any given fixed degree of activity, or activity greater or smaller than a fixed degree.