Estimation for linear parabolic SPDEs in two space dimensions with unknown damping parameters
Yozo Tonaki, Yusuke Kaino, Masayuki Uchida
TL;DR
The paper develops a two-stage estimation framework for a two-dimensional linear parabolic SPDE driven by two $Q$-Wiener models with unknown damping. It first constructs a damping parameter estimator from temporal-spatial realized quadratic variations and then uses a minimum-contrast approach, with the estimated damping plugged in, to recover the SPDE coefficients via an approximate coordinate process. The main contributions are (i) an $\sqrt{mN}$-rate estimator for the damping parameter, (ii) $\sqrt{mN}/\log N$-rate estimators for the coefficient parameters under unknown $\alpha$, and (iii) parallel results for both $Q_1$ and $Q_2$ models including asymptotic normality with explicit variance structures. Simulation studies illustrate small biases and favorable finite-sample performance, validating the proposed approach for high-frequency spatio-temporal data in 2D SPDEs. The work extends parametric SPDE estimation by accommodating unknown damping and providing rigorous thinning-based asymptotics, with potential applications in geophysical and environmental modeling.
Abstract
We study parametric estimation for second order linear parabolic stochastic partial differential equations (SPDEs) in two space dimensions driven by two types of $Q$-Wiener processes based on high frequency spatio-temporal data. First, we give estimators for damping parameters of the $Q$-Wiener processes of the SPDE using realized quadratic variations based on temporal and spatial increments. We next propose minimum contrast estimators of four coefficient parameters in the SPDE and obtain estimators of the rest of unknown parameters in the SPDE using an approximate coordinate process. We also examine numerical simulations of the proposed estimators.