Information bounds for inference in stochastic evolution equations observed under noise

TL;DR

This paper establishes fundamental limits on the accuracy of estimating parameters in stochastic evolution equations, especially SPDEs, under noisy observations, revealing how noise, observation time, and system properties influence estimation efficiency.

Contribution

It derives lower bounds on estimation errors for parameters in stochastic PDEs and demonstrates that a general estimation method can achieve these bounds, confirming their minimax optimality.

Findings

Lower bounds depend on noise level, observation time, and system properties.

A general estimation procedure attains the minimax bounds in many cases.

Nonparametric bounds reveal complex information structures.

Abstract

We consider statistics for stochastic evolution equations in Hilbert space with emphasis on stochastic partial differential equations (SPDEs). We observe a solution process under additional measurement errors and want to estimate a real or functional parameter in the drift. Main targets of estimation are the diffusivity, transport or source coefficient in a parabolic SPDE. By bounding the Hellinger distance between observation laws under different parameters we derive lower bounds on the estimation error, which reveal the underlying information structure. The estimation rates depend on the measurement noise level, the observation time, the covariance of the dynamic noise, the dimension and the order, at which the parametrised coefficient appears in the differential operator. A general estimation procedure attains these rates in many parametric cases and proves their minimax optimality. For nonparametric estimation problems, where the parameter is an unknown function, the lower bounds exhibit an even more complex information structure. The proofs are to a large extent based on functional calculus, perturbation theory and monotonicity of the semigroup generators.