Kolmogorov bounds for drift parameter estimation of continuously-observed SPDEs

Abstract

The purpose of this paper is to study the asymptotic behavior of the maximum likelihood estimator (MLE) and the minimum contrast estimator (MCE) of the drift coefficient for a stochastic partial differential equation based on continuous time observations of the Fourier coefficients $u_k\left(t\right),k=1,\dots ,N$ of the solution, over some finite interval of time $[0,T]$. More precisely, we derive Berry–Esseen bounds in Kolmogorov distance for the MLE and MCE when $N\to \infty$ and/or $T\to \infty$. Moreover, we prove the strong consistency of the MCE as $N\to \infty$ and/or $T\to \infty$.