Non‐parametric estimation for pure jump irregularly sampled or noisy Lévy processes

Abstract

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.