Asymptotic Properties of the Maximum Likelihood Estimator for Stochastic Parabolic Equations with Additive Fractional Brownian Motion

TL;DR

This paper investigates the asymptotic behavior of the maximum likelihood estimator for a stochastic parabolic equation driven by fractional Brownian motion, providing conditions for its consistency and normality as data size grows.

Contribution

It introduces new conditions based on eigenvalues for the estimator's consistency and asymptotic normality in fractional stochastic PDEs.

Findings

Derived necessary and sufficient conditions for estimator consistency.

Established asymptotic normality criteria based on eigenvalues.

Analyzed the impact of fractional Brownian motion on estimation properties.

Abstract

A parameter estimation problem is considered for a diagonaliazable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter $H\geq 1/2$. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation.