In this paper, we study the kernel estimation of the instantaneous coefficient of the semi-martingales, where the underlying process can contain a jump part of infinite variation. The estimator is based on the representation of the characteristic function of the Levy processes. The consistency of the estimator is established for Ito semi-martingales. By additionally assuming that the process behaves like a stable Levy process within a neighborhood of zero, the central limit theorem is established, which shows that the proposed estimator is variance efficient. Simulation studies justify the theoretical results, and we also apply the estimator to some real high-frequency database.
Dr. Zhi LIU is now an assistant professor in the Department of Mathematics, University of Macau. He obtained his Ph.D. degree in Statistics in Hong Kong University of Science and Technology, before he joined the University of Macau, Dr. Zhi LIU held a joint position of assistant professor in Economics in WISE and in Statistics in Department of Statistics, Xiamen University, China. His current research interests include statistical inference of stochastic processes, high dimensional data analysis, etc. He has published about 30 articles in journals of different areas, such as Annals of Statistics, Journal of American Statistical Association, Bioinformatics, Journal of Econometrics, Econometric Theory, Journal of Business and Economic Statistics, Finance and Stochastics, etc.