Vector-valued Generalised Ornstein-Uhlenbeck Processes

Abstract

Generalisations of the Ornstein-Uhlenbeck process defined through Langevin equation $dU_t = - \Theta U_t dt + dG_t,$ such as fractional Ornstein-Uhlenbeck processes, have recently received a lot of attention in the literature. In particular, estimation of the unknown parameter $\Theta$ is widely studied under Gaussian stationary increment noise $G$. Langevin equation is well-known for its connections to physics. In addition to that, motivation for studying Langevin equation with a general noise $G$ stems from the fact that the equation characterises all univariate stationary processes. Most of the literature on the topic focuses on the one-dimensional case with Gaussian noise $G$. In this article, we consider estimation of the unknown model parameter in the multidimensional version of the Langevin equation, where the parameter $\Theta$ is a matrix and $G$ is a general, not necessarily Gaussian, vector-valued process with stationary increments. Based on algebraic Riccati equations, we construct an estimator for the matrix $\Theta$. 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 characterises all stationary processes in a multidimensional setting as well.