扩散过程是随机微分方程中最为重要的模型之一，有着广阔的应用背景，特别 是在金融数学领域。有关扩散过程的统计推断已经成为最近二三十年来一个活跃的研究领 域。基于连续时间观测值的研究成果相对比较成熟，但在实际应用中，所得到的观测值都是 在离散时间下进行抽样的。本项目拟沿着扩散过程离散化的思路，研究扩散过程若干统计大 偏差问题。具体内容包括：（1）通过研究扩散过程局部时的大偏差性质，得到扩散系数核密 度估计的大偏差原理；（2）发展新的似然函数逼近方法，研究在抽样时间间隔小（大）的情 形下，极大似然估计的大偏差原理；（3）在非随机抽样和随机抽样的情形下，研究扩散过程 未知参数的极小对比估计，得到其大偏差原理。

Diffusion processes is one of the most important models in stochastic differential equations, which has wide application backgrounds, especially in mathematical finance. Statistical inference for diffusion processes has been an active research area during the last two or three decades. The research results are relatively mature based on the continuous-time observations, but in practical applications, data are essentially always recorded at discrete points in time only. Along with the idea of discretization of diffusion processes, this project mainly discusses the large deviations for some statistical problems of diffusion processes. (1) Based on the large deviation of local time, the large deviation of the kernel density estimator of the coefficient of a diffusion is established. (2) We want to develop some new methods to approximate maximum likelihood function, and obtain the large deviation of maximum likelihood estimator for the cases of different time interval. (3) We consider the nonrandom or random sampling schemes, and study the large deviation principle of the minimum contrast estimator of the unknown parameter for diffusion processes.