Statistical inference for a stochastic wave equation with Malliavin calculus

TL;DR

This paper investigates the asymptotic behavior of the maximum likelihood estimator for the wave speed in a stochastic wave equation, employing spectral methods, stochastic analysis, and Malliavin calculus, supported by numerical experiments.

Contribution

It introduces a spectral approach to derive and analyze the MLE for the wave speed, proving its consistency and asymptotic normality using advanced stochastic techniques.

Findings

Proved the consistency of the MLE for the wave speed.

Established the asymptotic normality of the MLE using Malliavin-Stein method.

Validated theoretical results with numerical simulations.

Abstract

In this paper we study asymptotic properties of the maximum likelihood estimator (MLE) for the speed of a stochastic wave equation. We follow a well-known spectral approach to write the solution as a Fourier series, then we project the solution to a $N$-finite dimensional space and find the estimator as a function of the time and $N$. We then show consistency of the MLE using classical stochastic analysis. Afterward we prove the asymptotic normality using the Malliavin-Stein method. We also study asymptotic properties of a discretized version of the MLE for the parameter. We provide this asymptotic analysis of the proposed estimator as the number of Fourier modes, $N$, used in the estimation and the observation time go to infinity. Finally, we illustrate the theoretical results with some numerical experiments.