Statistical Inference And Applications For Diffusion Coefficient Of Diffusion Models By Nonparametric Approach

This dissertation primarily studies the problems of statistical inference and applications for volatility of diffusion models by non-parametric approach, including the estimation of diffusion coefficient of diffusion models based on high-frequency data, the estimation of diffusion coeffi-cient of diffusion models based on high-frequency noisy data, threshold bipower estimation of spot volatility for jump-diffusion models, two-step estimation of diffusion processes using noisy observations. The main contents are arranged as follows:Chapter 1 is devoted to introducing the research background, significance, status and the existing problems for diffusion models. In addition, an outline of this dissertation about our major work is given, encompassing innovative keys.Chapter 2 focuses on the estimation of diffusion coefficient of diffusion models based on high-frequency data. By the use of low-frequency data and high-frequency data, a two-step estimation method is proposed to estimate the diffusion coefficient, which can robustly estimate it without knowing the drift term. Under some mild conditions, the asymptomatic properties of the resulting estimators can be established. Furthermore, several numerical studies axe con-ducted to evaluate the finite sample performance of the proposed methodologies. Finally, an application with real data illustrates the usefulness of the proposed techniques.Chapter 3 is devoted to studying the estimation of diffusion coefficient of diffusion models based on high-frequency noisy data. In practical applications, using intraday high-frequency data to implement directly estimate diffusion coefficient could be misleading, because intraday high-frequency data display microstructure effects that could seriously distort the estimate, and the unrobustness caused by the microstructure noise increases as the sample size. Thus, how to us intraday high-frequency data in the diffusion coefficient estimation is important. An alternative method is proposed based on the two-scales realized volatility approach. Others are also proposed, which can filter out the noise. Further, the practical problem of implementation for procedure parameters is discussed. Under some mild conditions, the asymptotic properties of the proposed estimators are established. Finally, results of two Monte Carlo experiments are presented to examine the finite sample performances of the proposed procedures and one empirical examples are discussed.Chapter 4 is concerned with the threshold bipower estimation of spot volatility for diffusion models with jumps. The newly defined estimator is based on the joint use of realized bipower variation and the threshold technique, and is not only consistent, but also scarcely plagued by small sample bias. The proposed estimator is shown to be robust to the presence of jumps and quite inelastic with respect to the choice of the threshold function, and retains the advantages of realized bipower variation and threshold technique. Under some mild conditions, the consistency and asymptotic normality for these estimators are studied explicitly. Moreover, some simulation studies are carried out to examine the finite sample performance of the proposed methods.Finally, the methodologies are illustrated by a real data set.Chapter 5 investigates the problem of the two-step estimation of diffusion processes using noisy observations. In order to reduce the noise effect, a two-step estimation method is proposed,which is based on the joint use of the pre-averaging technique and kernel smoothing. Under some suitable conditions, we show that the proposed estimators are consistent and asymptotically normal. Further, the finite sample performance of the proposed procedures is assessed by Monte Carlo simulation experiments. Finally, the proposed methodology is illustrated with an analysis of a real data set example.