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 $u_k(t)$ $u_{k}(t)$ , $k=1,\dots ,N$ $k=1, \ldots , N$ over a finite interval of time $[0,T]$ $[0, T]$ are observed on a uniform time grid: $0=t_0<t_1<\cdots <t_M=T$ $0=t_{0}< t_{1}<\cdots <t_{M}=T$ , with $\mathrm{\Delta }:=t_i-t_{i-1}=T/M$ $\Delta :=t_{i}-t_{i-1}={T}/{M}$ , $i\dot{}=1,\dots ,M$ $i\dot{}=1, \ldots , M$ . We provide an explicit Berry–Esseen bound in Kolmogorov distance for this approximative MLE when $N,M,T\to \mathrm{\infty }$ $N, M, T \rightarrow \infty $ , assuming that $T^3{N}^7/M^2\to 0$ ${T^{3}N^{7}}/{M^{2}}\rightarrow 0$ and $N^2/T\to 0$ $N^{2}/T\rightarrow 0$ .