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

Abstract

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.