Statistical inference for the stochastic wave equation based on discrete observations

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 Fejér-type kernels, enabling a precise analysis of the interplay between spatial and temporal contributions.

💡 Research Summary

The paper addresses the statistical inference problem for the linear stochastic wave equation on an unbounded multidimensional spatial domain, driven by a Riesz‑type spatially coloured noise that is white in time. The unknown parameter of interest is the wave speed ϑ > 0. The authors propose a method‑of‑moments framework based on discrete observations of the solution field in space, in time, and jointly in space‑time.

Key to the approach is the use of second‑order increments (also called second‑order differences) of the observed field. For a fixed time t the spatial increments are
(I^{\text{sp}}k = u(t,x{k+1}) + u(t,x_{k-1}) - 2u(t,x_k))
with a regular spatial mesh (x_k = λkρ). Analogously, for a fixed spatial location x the temporal increments are
(I^{\text{te}}i = u(t{i+1},x) + u(t_{i-1},x) - 2u(t_i,x))
with a regular temporal mesh (t_i = δi). The realised quadratic variations are defined as the sums of squared increments, (V_{\text{sp}} = \sum_k I^{\text{sp}}k{}^2) and (V{\text{te}} = \sum_i I^{\text{te}}i{}^2). A mixed space‑time variation (V{\text{sp,te}}) is built from box‑type increments that combine both directions.

The covariance structure of these increments can be expressed explicitly using Fourier analysis. The Fourier transform of the wave kernel is
(\mathcal F(G_t)(ω) = \sin(t\sqrt{ϑ}|ω|)/(\sqrt{ϑ}|ω|) = t,\operatorname{sinc}(t\sqrt{ϑ}|ω|)).
Consequently, the covariance of the solution takes the form
\