Estimation for SPDEs from noisy observations

We consider stochastic evolution equations of the form dX(t)=AθX(t)dt+BdWt

with a generator Aθ on a Hilbert space, involving an unknown real or functional parameter θ. We consider observations dY(t)=X(t)dt+εdVt, t∈[0,T], in space-time white noise dV and ask about optimal estimation of θ. Minimax lower bounds reveal a rich picture, which we shall describe in detail for second-order elliptic operators Aθ=∇⋅(θ2∇+θ1)+θ0. Optimal rates depend on the order of the coefficient θi, the dimension and the asymptotics taken. An even richer structure appears for nonparametric estimation. The lower bound proofs rely on Hellinger bounds for cylindrical Gaussian measures and functional calculus for non-commuting, unbounded normal operators. A rate-optimal parametric estimator is obtained by a subtle preaveraging approach. Finally, a nonparametric diffusivity estimator and several open problems are presented.