  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.
