Vector‐valued generalized Ornstein–Uhlenbeck processes: Properties and parameter estimation
研究了多维Langevin方程中未知参数矩阵的估计问题，其中噪声为一般向量值平稳增量过程，不限于高斯噪声。基于代数Riccati方程构造了估计量，证明了相合性并推导了极限分布。
Abstract 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.