Statistical inference for the stochastic wave equation based on discrete observations

TL;DR

This paper develops a method to estimate the wave speed in a stochastic wave equation using discrete data, establishing asymptotic normality and analyzing the covariance structure of the observations.

Contribution

It introduces a novel estimation approach for the wave speed in stochastic wave equations based on second-order variations and provides a detailed asymptotic analysis.

Findings

Method-of-moments estimators are asymptotically normal.

Closed-form covariance structure involving Fejér-type kernels.

Effective analysis of spatial and temporal sampling effects.

Abstract

The wave speed of a stochastic wave equation driven by Riesz noise on the unbounded multidimensional spatial domain is estimated based on discrete measurements. Central limit theorems for second-order variations of the observations in space, time, and space-time are established. Under general assumptions on the spatial and temporal sampling frequencies, the resulting method-of-moments estimators are asymptotically normally distributed. The covariance structure of the discrete increments admits a closed-form representation involving two different Fej\'er-type kernels, enabling a precise analysis of the interplay between spatial and temporal contributions.