Parameter estimation for SPDEs based on discrete observations in time and space

TL;DR

This paper develops statistical methods for estimating parameters in a stochastic PDE from discrete space-time data, proving their asymptotic properties and demonstrating their robustness and limitations.

Contribution

It introduces new estimators based on quadratic variations for SPDE parameters, with proven asymptotic normality and analysis of convergence rates.

Findings

Central limit theorems for quadratic variations in SPDEs

Robust method of moments estimators for diffusivity and volatility

Optimal convergence rates are generally slower than parametric rates

Abstract

Parameter estimation for a parabolic linear stochastic partial differential equation in one space dimension is studied observing the solution field on a discrete grid in a fixed bounded domain. Considering an infill asymptotic regime in both coordinates, we prove central limit theorems for realized quadratic variations based on temporal and spatial increments as well as on double increments in time and space. Resulting method of moments estimators for the diffusivity and the volatility parameter inherit the asymptotic normality and can be constructed robustly with respect to the sampling frequencies in time and space. Upper and lower bounds reveal that in general the optimal convergence rate for joint estimation of the parameters is slower than the usual parametric rate. The theoretical results are illustrated in a numerical example.