Drift estimation for jump diusions: time-continuous and high-frequency observations

The problem of parametric drift estimation for a Lévy-driven jump diffusion process is considered in two different settings: time-continuous and high-frequency observations. The goal is to develop explicit maximum likelihood estimators for both observation schemes that are efficient in the Hájek-Le Cam sense. In order to develop a maximum likelihood approach the absolute continuity and singularity problem for the induced measures on the path space is discussed. For varying drift parameter we obtain locally equivalent measures when the driving Lévy process has a Gaussian component. The likelihood function based on time-continuous observations can be derived explicitly and leads to explicit maximum likelihood estimators for several popular model classes. We consider Ornstein-Uhlenbeck type, square-root and linear stochastic delay differential equations driven by Lévy processes in detail and prove strong consistency, asymptotic normality and efficiency of the likelihood estimators in these models. The appearance of the continuous martingale part of the observed process under the dominating measure in the likelihood function leads to a jump filtering problem in this context, since the continuous part is usually not directly observable and can only be approximated and the high-frequency limit. This leads to the question how the jumps of the driving Lévy process influence the estimation error. We show that when the continuous part can only be recovered up to some small jumps the estimation error is proportional to the jump intensity of these small jumps. Hence, efficient jump filtering becomes an important task before inference on the drift can be undertaken. As a side result we obtain that least squares estimation is inefficient when jumps are present. In the second part of this thesis the problem of drift estimation for discretely observed processes is considered. The estimators are constructed from discretizations of the timecontinuous maximum likelihood estimators from the first part, where the continuous martingale part is approximated via a threshold technique. Here the jump activity of the Lévy process plays a crucial role for the asymptotic analysis of the estimators. We consider first the case of finite activity and show that under suitable conditions on the behavior of small jumps and the observation frequency the drift estimator attains the efficient asymptotic distribution that we have derived in the first part. Based on these results we prove asymptotics normality and efficiency for the drift estimator in the Ornstein Uhlenbeck type model also for infinite jump activity. In the course of the proof we show that the continuous part of a jump diffusion can be recovered in the high frequency limit even when the observation horizon growth to infinity and the process has infinitely many small jumps in every finite time interval. Finally, the finite sample behavior of the estimators is investigated on simulated data. When the assumption of high-frequency observations is reasonable the theoretical results are confirmed. We find also that the maximum likelihood approach clearly outperforms the least squares estimator when jumps are present and that the efficiency gap between both techniques becomes even more severe with growing jump intensity.