Statistical inference for a stochastic wave equation with Malliavin calculus

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.