Kolmogorov bounds for maximum likelihood drift estimation for discretely sampled SPDEs

In this paper, we investigate an approximative maximum likelihood estimator (MLE) for the drift coefficient of a stochastic partial differential equation in the case where the corresponding Fourier coefficients $u_{k}(t)$ , $k=1, \ldots , N$ over a finite interval of time $[0, T]$ are observed on a uniform time grid: $0=t_{0}< t_{1}<\cdots <t_{M}=T$ , with $\Delta :=t_{i}-t_{i-1}={T}/{M}$ , $i\dot{}=1, \ldots , M$ . We provide an explicit Berry–Esseen bound in Kolmogorov distance for this approximative MLE when $N, M, T \rightarrow \infty $ , assuming that ${T^{3}N^{7}}/{M^{2}}\rightarrow 0$ and $N^{2}/T\rightarrow 0$ .