Optimal Berry-Esseen bound for parameter estimation of SPDE with small noise

We investigate a rate of convergence on asymptotic normality of the maximum likelihood estimator (MLE) for parameter θ appearing in parabolic SPDEs of the form duϵ(t,x)=(A0+θA1)uϵ(t,x)dt+ϵdW(t,x),where A0 andA1 are partial differential operators, W is a cylindrical Brownian motion (CBM) and ϵ↓0. We find an optimal Berry-Esseen bound for central limit theorem (CLT) of the MLE. It is proved by developing techniques based on combining Malliavin calculus and Stein's method.