[Paper Review] Statistical inference for the stochastic wave equation based on discrete observations

Anton Tiepner, Mathias Trabs+1 more|arXiv (Cornell University)|Feb 4, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper develops central limit theorems for second-order space, time, and space-time variations of the stochastic wave equation driven by Riesz noise, enabling asymptotically normal method-of-moments estimators for the wave speed using discrete observations.

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

Motivation & Objective

• Motivate estimation of the wave speed parameter ϑ for a stochastic wave equation with spatially colored noise.
• Develop method-of-moments estimators based on second-order space, time, and space-time variations.
• Establish central limit theorems for these estimators under general spatial and temporal sampling schemes.
• Characterize covariance structures through Fejér-type kernels to analyze space-time interactions.

Proposed method

• Model the stochastic wave equation with Riesz noise and known β on Rd.
• Define second-order spatial variation Vsp using spatial increments Isp,k and derive its covariance via Fejér kernels.
• Derive a central limit theorem for √n of λβ−2/n Vsp with bias and variance constants Csp,E and Csp,V.
• Define second-order temporal variation Vte with temporal increments Ite,i and obtain a CLT for √m of δβ−3/m2 Vte with constants Cte,E and Cte,V.
• Develop space-time increments Vsp,te with Fejér-type representations and analyze its asymptotics in regimes α = δ/λ.
• Construct method-of-moments estimators for ϑ from Vsp and Vte and derive their asymptotic normality using the delta method.

Experimental results

Research questions

• RQ1 How can ϑ, the wave speed, be consistently estimated from discrete space-time observations of the stochastic wave equation?
• RQ2 What are the asymptotic distributions of second-order space, time, and space-time variations under general sampling frequencies?
• RQ3 How do Fejér-type kernels describe the covariance structure of discrete increments and how do they influence CLTs?
• RQ4 Can method-of-moments estimators based on these variations achieve asymptotic normality and optimal rates?

Key findings

• Second-order spatial variations yield an asymptotically normal estimator for ϑ with rate √n and bias controlled by λ; the limit variance involves a constant Csp,V and the limit requires λ→0.
• Second-order temporal variations yield an asymptotically normal estimator for ϑ with rate √m and variance involving Cte,V; the limit holds under m→∞.
• Space-time second-order variations lead to CLTs with rates depending on the hyperbolic sampling ratio α, capturing dominance of spatial or temporal contributions in different regimes.
• The covariance structure of increments admits closed-form Fejér-kernel representations enabling precise analysis of space-time interplay.
• Delta-method-based estimators from Vsp and Vte provide asymptotically normal ϑ estimates with explicit variance constants and enable confidence interval construction.