论文标题
由普通高斯噪声驱动的Ornstein-Uhlenbeck过程的参数估计
Parameter estimation for an Ornstein-Uhlenbeck Process driven by a general Gaussian noise
论文作者
论文摘要
在本文中，我们考虑了由一般一维中心的高斯过程$（g_t）_ {t \ ge 0} $驱动的Ornstein-Uhlenbeck过程的推论问题。协方差函数$ r（t，\，s）= \ mathbb {e} [g_t g_s] $的第二阶混合部分导数可以分解为两个部分，其中一个部分与$（ts）^{β-1} $（β-1} $相一致。此条件对于一类连续的高斯流程有效，这些过程无法自相似或具有固定的增量。一些例子包括布朗运动次数和双分裂的布朗运动。在此假设下，我们研究了由高斯噪声$（g_t）_ {t \ ge 0} $驱动的Ornstein-uhlenbeck过程中漂移参数的参数估计。对于最小二乘估计器和第二次估计量，我们从连续的观测值中构成，我们证明了强的一致性和不良性正态性，并获得了浆果 - 埃塞恩界限。证明是基于内部产品的Hilbert Space $ \ Mathfrak {H} $与高斯噪声$（G_T）_ {T \ GE 0} $相关的，以及基于与分数Brownian运动相关的Hilbert Space的结果的估计。
In this paper, we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{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 $(ts)^{β-1}$ up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or have stationary increments. Some examples include the subfractional Brownian motion and the bi-fractional Brownian motion. Under this assumption, we study the parameter estimation for drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise $(G_t)_{t\ge 0}$. 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 $(G_t)_{t\ge 0}$, and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.
