Nonparametric estimation for a class of Lévy processes

We consider estimation for a class of Lévy processes, modelled as a sum of a drift, a symmetric stable process and a compound Poisson process. We propose a nonparametric approach to estimating unknown parameters of our model, including the drift, the scale and index parameters in the stable law, the mean of the Poisson process and the underlying jump size distribution. We show that regression and nonparametric deconvolution methods, based on the empirical characteristic function, can be used for inference. Interesting connections are shown to exist between properties of our estimators and of those found in conventional deconvolution.