[Paper Review] Parameter Estimation in Diagonalizable Stochastic Hyperbolic Equations

W. Liu, Sergey V. Lototsky|ArXiv.org|Jun 23, 2009

Stochastic processes and financial applications9 references3 citations

TL;DR

This paper develops a spectral-type maximum likelihood estimator for unknown parameters in a diagonalizable stochastic hyperbolic equation driven by space-time white noise, where the damping and stiffness operators are unbounded. As the number of Fourier modes increases, the estimator achieves consistency and asymptotic normality under mild conditions on the eigenvalues, establishing theoretical validity for parameter inference in infinite-dimensional stochastic PDEs with additive noise.

ABSTRACT

A parameter estimation problem is considered for a linear stochastic hyperbolic equation driven by additive space-time Gaussian white noise. The damping/amplification operator is allowed to be unbounded. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed.

Motivation & Objective

• To establish conditions under which a linear stochastic hyperbolic equation with unbounded damping and stiffness operators admits a generalized solution in a Hilbert space.
• To construct a maximum likelihood estimator for two unknown parameters using finite-dimensional projections of the solution's Fourier coefficients.
• To analyze the asymptotic properties—consistency and asymptotic normality—of the estimator as the number of Fourier modes increases.
• To extend parameter estimation theory from stochastic parabolic to hyperbolic equations, particularly in the context of diagonalizable, second-order SPDEs with additive noise.

Proposed method

• The solution is represented as a Fourier series in an orthonormal basis of eigenfunctions common to the operators A₀, A₁, B₀, and B₁, reducing the PDE to an infinite system of uncoupled SDEs.
• The maximum likelihood estimator for (θ₁, θ₂) is derived in closed form using observed trajectories of the first N Fourier coefficients and their time derivatives.
• Hyperbolicity is ensured by requiring the sequence (κₖ + θτₖ + C*) to be positive, non-decreasing, and unbounded for all θ in the parameter space.
• Consistency and asymptotic normality are proven using a martingale central limit theorem applied to normalized sums of stochastic integrals over Fourier modes.
• The analysis relies on conditions (A.11)–(A.13) on the eigenvalue sequences, with (A.13) being sufficient for weak laws and (A.11)–(A.12) for strong laws.
• Theoretical results are established under the assumption of zero initial conditions and Hilbert-Schmidt embedding of the solution space into a larger Hilbert space X.

Experimental results

Research questions

• RQ1 Under what conditions does a stochastic hyperbolic equation with unbounded operators admit a generalized solution in a Hilbert space?
• RQ2 Can a maximum likelihood estimator be constructed using only a finite number of Fourier coefficients of the solution?
• RQ3 Does the estimator for the two unknown parameters (θ₁, θ₂) remain consistent and asymptotically normal as the number of Fourier modes increases?
• RQ4 What eigenvalue growth conditions are necessary and sufficient for the consistency and asymptotic normality of the spectral estimator?

Key findings

• The estimator for (θ₁, θ₂) is consistent as the number of Fourier modes N → ∞, provided the eigenvalue sequences satisfy the slowly increasing condition (A.13).
• Asymptotic normality of the estimator is established under stronger conditions (A.11) and (A.12), which imply (A.13), ensuring convergence in distribution to a bivariate normal distribution.
• The proof relies on a martingale central limit theorem applied to normalized stochastic integrals of the Fourier coefficients, with the normalization based on expected quadratic variation.
• The conditions (A.11) and (A.12) are shown to be stronger than (A.13), and their joint satisfaction ensures strong consistency of the estimator.
• For the special case of (pseudo)differential elliptic operators on a bounded domain, the theoretical results hold under standard boundary conditions and smoothness assumptions.
• The framework allows for unbounded damping and stiffness operators, extending parameter estimation theory beyond bounded or compact operators.