Optimal parameter estimation for linear SPDEs from multiple measurements

TL;DR

This paper develops a method for estimating coefficients in linear SPDEs from multiple measurements, achieving optimal convergence rates depending on the coefficients' differential order, with theoretical guarantees of consistency and minimax optimality.

Contribution

It introduces a novel estimation scheme based on reproducing kernel Hilbert space analysis, providing explicit convergence rates and conditions for consistent estimation of SPDE coefficients.

Findings

Convergence rates depend on coefficient order, faster for higher order.

The estimation scheme is minimax optimal.

Conditions for consistent estimation are established.

Abstract

The coefficients in a second order parabolic linear stochastic partial differential equation (SPDE) are estimated from multiple spatially localised measurements. Assuming that the spatial resolution tends to zero and the number of measurements is non-decreasing, the rate of convergence for each coefficient depends on its differential order and is faster for higher order coefficients. Based on an explicit analysis of the reproducing kernel Hilbert space of a general stochastic evolution equation, a Gaussian lower bound scheme is introduced. As a result, minimax optimality of the rates as well as sufficient and necessary conditions for consistent estimation are established.