Estimation of the mean of stationary and nonstationary Ornstein-Uhlenbeck processes and sheets

We consider the problem of estimating an unknown parameter m in case one observes in an interval (rectangle) stationary and nonstationary Ornstein-Uhlenbeck processes (sheets), which are shifted by m times a known deterministic function on the interval (rectangle). It turns out that the maximum likelihood estimator (MLE) has a normal distribution and, for instance, in case of the sheet this MLE is a weighted linear combination of the values at the vertices, integrals on the edges, and the integral on the whole rectangle of the weighted observed process. We do not use partial stochastic differential equations; we apply direct discrete time approach instead. To make the transition from the discrete time to the continuous time, a tool is developed, which might be of independent interest.