Functional estimation for Lévy measures of semimartingales with Poissonian jumps

We consider semimartingales with jumps that have finite Lévy measures. This paper is devoted to the estimation of integral-type functionals of the Lévy measures, which are sometimes functions on a compact space, and our main interest is the inference from discrete observations. However we first consider the case where continuous observations are obtained. It would give us an important insight to the discretely observed case, that is, estimators of our target functions by discrete observations are constructed as a natural discretization of ones by continuous observations. We show the L2-consistency, the uniform consistency on compacts and the asymptotic normality with the minimal asymptotic variance of our estimators.