Jump filtering and efficient drift estimation for Lévy-Driven SDE’S

The problem of efficient drift estimation for a parametric class of solutions of stochastic differential equations with Lévy-type jumps is considered under discrete high-frequency observations with growing observation window. The main challenge in this estimation problem stems from the two very different sources of noise involved: continuous diffusion and jump component of the process. This is re ected by the appearance of the unobserved continuous martingale part in the likelihood function. In order to obtain a feasible and effcient drift estimator based on discrete observations a jump filtering technique is employed to obtain a nonparametric estimators of integrals with respect to the continuous part. We prove general convergence results for these nonparametric estimators that are essential in any estimation problem concerning the continuous part such as drift an volatility estimation. Based on an LAN result for the general model this enables us finally to prove asymptotic e ciency in the sense of Há jek-Le Cam for the resulting drift estimator with jump filter. We then illustrate consequences of this general theory for a number of specific jump diffusion models, including the Cox-Ingersoll-Ross model with jumps from finance or the class of Ornstein-Uhlenbeck type processes. Another advantage of our approach are the straightforward implementation and the low computational costs which are demonstrated in a short simulation study that shows excellent agreement with our theoretical results.