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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.

Statistical inference for a stochastic wave equation with Malliavin–Stein method