Statistical inference for stochastic differential equations from discrete observation and its applications

从固定区间上观测到的高频数据出发，考虑随机微分方程波动参数的统计渐近推断。利用统计随机场的多项式型大偏差不等式，证明了波动率参数的极大似然型估计量和Bayes型估计量的渐近混合正态以及估计量的收敛矩.在nh^p-&gt；0下，其中h为离散化步长，n为样本量，p为大于2的整数值，得到了离散遍历扩散过程漂移和波动参数的自适应极大似然估计，并证明了它们的渐近正态和矩收敛.此外，我们还研究了离散观测中错误指定的遍历扩散过程的漂移和波动参数的自适应估计，并证明了它们的渐近性质。

We considered statistically asymptotic inference for volatility parameters of stochastic differential equations from high frequency data observed on the fixed interval. By using the polynomial type large deviation inequality for the statistical random field, we showed the asymptotic mixed normality of maximum likelihood type estimator and Bayes type estimator of the volatility parameter and the moments of convergences of the estimators. Under nh^p ->0, where h is the discretization step size, n is the sample size and p is an integer value greater than 2, adaptive maximum likelihood estimators of both drift and volatility parameters for discretely observed ergodic diffusion processes were obtained, and their asymptotic normality and moment convergence wereproved. Moreover, we treated adaptive estimators of drift and volatility parameters for misspecified ergodic diffusion processes from discrete observations and their asymptotic properties were shown.