Drift Estimation for Discretely Sampled SPDEs

TL;DR

This paper investigates the asymptotic behavior of maximum likelihood estimators for the drift coefficient in fractional stochastic heat equations, considering both spectral domain measurements and discretized observations over time.

Contribution

It provides a comprehensive analysis of the asymptotic properties of MLEs for SPDEs under both continuous and discretized sampling schemes, including conditions for consistency and normality.

Findings

MLE is consistent and asymptotically normal in continuous observation setting.

Discretized MLEs are consistent and normal under specific growth conditions of N, M, and T.

The paper establishes optimality and asymptotic properties of estimators in spectral domain SPDEs.

Abstract

The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature, we study the asymptotic properties of the maximum likelihood (type) estimators (MLE) when both, the number of Fourier modes and the time go to infinity. In the first part of the paper we consider the usual setup of continuous time observations of the Fourier coefficients of the solutions, and show that the MLE is consistent, asymptotically normal and optimal in the mean-square sense. In the second part of the paper we investigate the natural time discretization of the MLE, by assuming that the first N Fourier modes are measured at M time grid points, uniformly spaced over the time interval [0,T]. We provide a rigorous asymptotic analysis of the proposed estimators when N goes to infinity and/or T, M go to infinity. We establish sufficient conditions on the growth rates of N, M and T, that guarantee consistency and asymptotic normality of these estimators.