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Parameter Estimation for an Ornstein-Uhlenbeck Process Driven by a General Gaussian Noise
Acta Mathematica Scientia ( IF 1 ) Pub Date : 2021-01-29 , DOI: 10.1007/s10473-021-0218-x
Yong Chen , Hongjuan Zhou

In this paper, we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process (Gt)t≥0. The second order mixed partial derivative of the covariance function \(R(t,s) = \mathbb{E}\left[ {{G_t}{G_s}} \right]\) can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by (ts)β−1 up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or to have stationary increments; some examples of this include the subfractional Brownian motion and the bi-fractional Brownian motion. Under this assumption, we study the parameter estimation for a drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise (Gt)t≥0. For the least squares estimator and the second moment estimator constructed from the continuous observations, we prove the strong consistency and the asympotic normality, and obtain the Berry-Esséen bounds. The proof is based on the inner product’s representation of the Hilbert space \(\mathfrak{h}\) associated with the Gaussian noise (Gt)t≥0, and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.



中文翻译:

一般高斯噪声驱动的Ornstein-Uhlenbeck过程的参数估计

在本文中,我们考虑一个奥恩斯坦-Uhlenbeck过程通过通常的一维驱动的推理问题居中高斯过程(ģ≥0。协方差函数\(R(t,s)= \ mathbb {E} \ left [{{G_t} {G_s}} \ right] \)的二阶混合偏导数可以分解为两部分,其中之一与分数布朗运动的重合,另一个由(tsβ -1限制直到一个恒定的因素。这个条件对于一类连续的高斯过程是有效的,这些过程不能自相似或具有平稳的增量。这样的一些例子包括亚分数布朗运动和双分数布朗运动。在这种假设下,我们研究了在由高斯噪声(驱动奥恩斯坦-Uhlenbeck过程的漂移参数的参数估计值G ^≥0。对于通过连续观测构造的最小二乘估计和第二矩估计,我们证明了强一致性和渐近正态性,并获得了Berry-Esséen界。该证明基于希尔伯特空间\(\ mathfrak {h} \)的内积表示与高斯噪声(相关ģ≥0,且内积的基于与所述分数布朗运动相关联的Hilbert空间的结果的估计。

更新日期:2021-01-29
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