Asymptotic theory for estimating the parameters of a Lévy process

We consider consistency and asymptotic normality of maximum likelihood estimators (MLE) for parameters of a Lévy process of the discontinuous type. The MLE are based on a single realization of the process on a given interval [0, t]. Depending on properties of the Lévy measure we either consider the MLE corresponding to jumps of size greater than ε and, keeping t fixed, we let ε tend to 0, or we consider the MLE corresponding to the complete information of the realization of the process on [0, t] and let t tend to ∞. The results of this paper improve in both generality and rigor previous asymptotic estimation results for such processes.