Nonparametric adaptive estimation for pure jump Levy processes

摘要

This paper is concerned with nonparametric estimation of the Levy density of a pure jump Levy process. The sample path is observed at n discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvo-lution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the L~2-risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the Levy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed.