Optimal Pointwise Approximation of Infinite-Dimensional Ornstein–uhlenbeck Processes

We consider an infinite-dimensional Ornstein–Uhlenbeck process on the spatial domain ]0,1[ d driven by an additive nuclear or space–time white noise, and we study the approximation of this process at a fixed point in time. We determine the order of the minimal errors as well as asymptotically optimal algorithms, both of which depend on the spatial dimension d and on the decay of the eigenvalues of the driving Wiener process W in the case of nuclear noise. In particular, the optimal order is achieved by employing drift-implicit Euler schemes with non-uniform time discretizations, while uniform time discretizations turn out to be suboptimal in general. By means of non-asymptotic error bounds and by simulation experiments, we show that the asymptotic results are predictive for the actual errors already for time discretizations with a small number of points.