Statistical Inference For Some Diffusion-type Processes

Diffusion type processes are a large class of continuous time processes which are widely used for stochastic modeling. For example, these processes are widely used for model building in the social, physical, engineering, and life sciences as well as in financial economics. Statistical inference for diffusion type processes is of great importance from the theoretical as well as from applications point of view in model building. This paper aims to study the statistical inference for three types of diffusion type processes based on high frequency sampling.The first part of this paper concerns integrated diffusion process, wrhich can not only model integrated and differentiated diffusion processes but also overcome the difficulties associated with the nondifferentiability of the Brownian motion. In this chapter, we propose a re-weighted estimator of the diffusion coefficient in the integrated diffusion model, the new estimator preserves the appealing bias properties of the local linear estimator and is guaranteed to be nonnegative in finite samples. Consistence of the estimator is proved under appropriate condi-tions and the conditions that ensure the asymptotie normality are also stated. The performance of the proposed estimators is assessed by simulation study.The second part of this paper concerns the statistical inference for drift and diffusion functions in integrated diffusion model. The empirical likelihood based estimators and the non-symmetric confidence intervals based on empirical like-lihood method for drift and diffusion functions in integrated diffusion process are constructed, and by comparing the non-symmetric confidence intervals with the symmetric ones based on normal approximation, the paper obtains the dif-ferences and quality of them. The consistency and asymptotic normality of the empirical likelihood estimators for drift and diffusion functions are obtained, and an adjusted empirical log-likelihood ratio is proved to be asymptotically standard chi-square under some mild conditions.The third part of this paper develops the empirical likelihood goodness-of-fit test for integrated diffusion process. This chapter uses the empirical likelihood technique to construct test statistic for the goodness-of-fit of a integrated diffu-sion model, this is because the empirical likelihood method has two attractive features. One is its automatic consideration of the variation associated with the nonparametric fit due to the empirical likelihood’s ability to studentize in-ternally. The other one is the asymptotic distributions of the test statistic are free of unknown parameters which avoids secondary plug-in estimation. More-over, the chapter discusses the asymptotic distribution of the test statistic, and investigates the testing procedure in simulations study.The fourth part of this paper develops an local M-estimation for drift and diffusion functions based on discrete-time observations of a diffusion process, the new estimators inherit the advantages of local linear smoother and over comes the shortcoming of lack of robustness of least-squares estimator. The consistency and asymptotic normality of the local M-estimators for drift and diffusion functions are obtained under mild conditions. The simulation studies demonstrate that the proposed estimators perform well in robustness.The last part of this paper concerns the diffusion process with jumps, the local M-estimators for the infinitesimal moments in the jump-diffusion model are obtained by combining of the local linear smoothing technique and the M-estimation technique. The consistency and asymptotic normality of the local M-estimators for the infinitesimal moments are obtained under mild conditions. The simulation studies demonstrate that the proposed estimators perform well in robustness.This research is supported by National Natural Science Foundation of China (No.11071214), Natural Science Foundation of Zhejiang Province (No. R6100119) and New Century Excellent Talents in University (NCET-08-0481).