Inference on the Lévy measure in case of noisy observations

Abstract

In this paper we are concerned with inference on the Lévy measure of a Lévy process in case of noisy high frequency observations. It is known that standard techniques for denoising, developed for diffusion settings, do not work in this case. For this reason, we provide an extension of the pre-averaging method which allows for a consistent estimation of the Lévy distribution function even under microstructure noise. Interestingly, the asymptotic behaviour of the novel estimator is the same as in the no-noise case. This is in sharp contrast to what is known for diffusions.