Trajectory Fitting Estimators for SPDEs Driven by Additive Noise

TL;DR

This paper introduces trajectory fitting estimators for linear parabolic SPDEs driven by additive noise, demonstrating their consistency and asymptotic normality when observing spectral modes.

Contribution

It proposes a novel spectral-based estimator for SPDE parameters, extending classical least squares methods to infinite-dimensional stochastic systems.

Findings

Estimator is consistent as the number of observed modes increases.

Estimator exhibits asymptotic normality under spectral observations.

Applicable to a broad class of linear parabolic SPDEs.

Abstract

In this paper we study the problem of estimating the drift/viscosity coefficient for a large class of linear, parabolic stochastic partial differential equations (SPDEs) driven by an additive space-time noise. We propose a new class of estimators, called trajectory fitting estimators (TFEs). The estimators are constructed by fitting the observed trajectory with an artificial one, and can be viewed as an analog to the classical least squares estimators from the time-series analysis. As in the existing literature on statistical inference for SPDEs, we take a spectral approach, and assume that we observe the first $N$ Fourier modes of the solution, and we study the consistency and the asymptotic normality of the TFE, as $N\to\infty$.