• adaptive nonparametric estimation;
  • Lévy processes;
  • high frequency–low frequency data;
  • irregular sampling;
  • noisy observations

In this paper, we study non-parametric estimation of the Lévy density for pure jump Lévy processes. We consider n discrete time observations that may be irregularly sampled or possibly corrupted by a small noise independent of the main process. The case of non-noisy observations with regular sampling interval has been studied by the authors in previous works which are the benchmark for the extensions proposed here. We study first the case of a regular sampling interval and noisy data, then the case of irregular sampling for non-noisy data. In each case, non adaptive and adaptive estimators are proposed and risk bounds are derived.