一类分数高斯噪声驱动的Ornstein-Uhlenbeck过程的参数估计:Hurst参数H∈(0,1/2)  

Parameter Estimation for an Ornstein-Uhlenbeck Process Driven by a Type of Gaussian Noise with Hurst Parameter H∈(0,1/2)

作　　者：陈勇[1] 李英 盛英 古象盟 Chen Yong;Li Ying;Sheng Ying;Gu Xiangmeng(School of Mathematics and Statistics,Jiangxi Normal University,Nanchang 330022;School of Mathematics and Computional Science,Xiangtan University,Xiangtan 411105)

机构地区：[1]江西师范大学数学与统计学院,南昌330022 [2]湘潭大学数学与计算科学学院,湖南湘潭411105

出　　处：《数学物理学报（A辑）》2023年第5期1483-1518,共36页Acta Mathematica Scientia

基　　金：国家自然科学基金(11961033,12171410);湖南省教育厅一般项目(22C0072)。

摘　　要：Chen和Zhou(2021)研究了一类分数高斯过程(G_(t))_(t≥0)驱动的Ornstein-Uhlenbeck过程的参数估计问题,其中协方差函数R(t,s)=E[G_(t)G_(s)]的二阶混合偏导分解成两个部分:一个与分数布朗运动相同,另一个以(ts)^(H−1)为界,其中H∈(1/2,1).该文研究同一问题,但假设H∈(0,1/2).分数高斯过程联系的希尔伯特空间H当H∈(1/2,1)和H∈(0,1/2)时差异显著.该文的起点是这类高斯过程(G_(t))t≥0和分数布朗运动(B^(H)_(t))t≥0分别联系的希尔伯特空间H和H_(1)的内积之间的一种定量关系.该文得到漂移参数基于连续时间观测的最小二乘估计和矩估计的强相合性,其中H∈(0,1/2),及渐近正态性和Berry-Esséen类上界,其中H∈(0,3/8).In 2021,Chen and Zhou consider an inference problem for an Ornstein-Uhlenbeck process driven by a type of centered fractional Gaussian process(G_(t))_(t≥0).The second order mixed partial derivative of the covariance function R(t,s)=E[G_(t)G_(s)]can be decomposed into two parts,one of which coincides with that of fractional Brownian motion and the other is bounded by(ts)^(H−1) with H∈(1/2,1),up to a constant factor.In this paper,we investigate the same problem but with the assumption of H∈(0,1/2).It is well known that there is a significant difference between the Hilbert space associated with the fractional Gaussian processes in the case of H∈(1/2,1)and that of H∈(0,1/2).The starting point of this paper is a quantitative relation between the inner product of H associated with the Gaussian process(G_(t))_(t≥0) and that of the Hilbert space H_(1) associated with the fractional Brownian motion(B^(H)_(t))t≥0.We prove the strong consistency with H∈(0,1/2),and the asymptotic normality and the Berry-Esséen bounds with H∈(0,3/8)for both the least squares estimator and the moment estimator of the drift parameter based on the continuous observations.

关 键 词：分数布朗运动 四阶矩定理 ORNSTEIN-UHLENBECK 过程 分数高斯过程 Berry-Esséen 类上界 

分 类 号：O211.64[理学—概率论与数理统计]