Parameter estimation for a linear parabolic SPDE model in two space dimensions with a small noise

TL;DR

This paper develops and analyzes estimators for parameters in a two-dimensional linear parabolic SPDE with small noise, using high-frequency spatial and temporal data, and addresses inference for the associated Ornstein-Uhlenbeck process.

Contribution

It introduces new minimum contrast estimators for SPDE parameters based on thinned high-frequency data in space and time, and proposes a drift estimator leveraging the Ornstein-Uhlenbeck structure.

Findings

Consistent estimators for SPDE coefficients are constructed.

The estimators perform well with high-frequency data in simulations.

The methodology extends to small noise diffusion inference.

Abstract

We study parameter estimation for a linear parabolic second-order stochastic partial differential equation (SPDE) in two space dimensions with a small dispersion parameter using high frequency data with respect to time and space. We set two types of $Q$-Wiener processes as a driving noise. We provide minimum contrast estimators of the coefficient parameters of the SPDE appearing in the coordinate process of the SPDE based on the thinned data in space, and approximate the coordinate process based on the thinned data in time. Moreover, we propose an estimator of the drift parameter using the fact that the coordinate process is the Ornstein-Uhlenbeck process and statistical inference for diffusion processes with a small noise.