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.

翻译：在本文中,我们研究了最大可能性估计值(MLE)对于随机波形方程式速度的无症状特性。我们采用了众所周知的光谱方法将溶液写成 Fourier 系列,然后我们将溶液投射为美元无限的维空间,发现估计值是时间的函数和美元。然后我们用古典的随机分析来显示 mLE的一致性。之后,我们用Malliavin-Stein 方法来证明无症状的正常性。我们还研究了参数的离散版本 MLE 的无症状特性。我们提供了对拟议估计值的估算值的无症状分析,即四倍模式的数值($N),即估算值和观察时间到无限的数值。最后,我们用一些数字实验来说明理论结果。