Generalizations of the Ornstein–Uhlenbeck process defined through Langevin equations, such as fractional Ornstein–Uhlenbeck processes, have recently received a lot of attention. However, most of the literature focuses on the one-dimensional case with Gaussian noise. In particular, estimation of the unknown parameter is widely studied under Gaussian stationary increment noise. In this article, we consider estimation of the unknown model parameter in the multidimensional version of the Langevin equation, where the parameter is a matrix and the noise is a general, not necessarily Gaussian, vector-valued process with stationary increments. Based on algebraic Riccati equations, we construct an estimator for the parameter matrix. Moreover, we prove the consistency of the estimator and derive its limiting distribution under natural assumptions. In addition, to motivate our work, we prove that the Langevin equation characterizes essentially all multidimensional stationary processes.

中文翻译：

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向量值广义 Ornstein-Uhlenbeck 过程：属性和参数估计

通过朗之万方程定义的 Ornstein-Uhlenbeck 过程的推广，例如分数 Ornstein-Uhlenbeck 过程，最近受到了很多关注。然而，大多数文献都集中在具有高斯噪声的一维情况。特别是，未知参数的估计在高斯平稳增量噪声下得到了广泛的研究。在本文中，我们考虑在多维版本的 Langevin 方程中估计未知模型参数，其中参数是矩阵，噪声是一般的、不一定是高斯的、具有固定增量的向量值过程。基于代数Riccati方程，我们构造了参数矩阵的估计器。此外，我们证明了估计量的一致性，并在自然假设下推导出了它的极限分布。