扩散型过程统计推断中极限理论的研究

Diffusion processes can be applied to physical、biological and medical sciences, and more recently to economic and social sciences. Diffusion processes usually involve unknown parameters which need to be estimated from observations on the processes. Estimating these parameters is an important topic. Recently, it is also one of hot issues in applied probability. This project, which will pay much attention to researches on limit theory of statistical inference for diffusion type processes, includes: . (1)from continuous observations, with the method of deviation inequality for quadratic functional and the transform of Girsanov, intending to research the complete moment convergence of the estimation of the unknown parameters for diffusion processes; . (2)from a discrete sampled data set for the process, with the method of the constructed deviation inequality for quadratic functional, intending to research the limit theory of the maximum likelihood estimation of the unknown parameters for diffusion, such as moderate deviations and deviation inequalities.

扩散过程可以广泛应用于物理、生物以及医学领域，尤其最近被应用于经济与社会科学。扩散过程中通常包含一些未知参数，对这些未知参数进行估计是一个重要研究内容，近年来，它也是应用概率论研究的热点之一。本课题将针对扩散型过程统计推断中的若干问题进行极限理论的研究，包括：. (1)在对轨道的连续观测下，利用二次泛函的偏差不等式、Girsanov 变换等工具来研究扩散过程中未知参数估计的完全矩收敛性质等; . (2)在对轨道的离散观测下，首先建立相应的二次泛函的偏差不等式等工具，然后研究扩散过程中未知参数极大似然估计的极限性质，例如中偏差原理以及偏差不等式等。

本项目基本按照原计划完成，并且主要取得如下的一些结果和进展：.（1）带一般漂移系数的反射Ornstein-Uhlenbeck扩散模型中未知参数极大似然估计的极限性质；. （2）反射Ornstein-Uhlenbeck扩散模型中未知参数极大似然估计的Cramer-Rao下界；. （3）带马氏切换的线性不确定随机系统的平稳性分析；. （4）多指标驱动的自正则和的完全矩的收敛速度；. （5）线性过程R/S统计量的渐近性质的一般模式。