Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach

TL;DR

This paper develops new central limit theorems for power variations of discretely sampled semilinear SPDEs, enabling improved estimation of model parameters and addressing previous conjectures about estimator bias.

Contribution

It introduces the concept of Δ-power variations, proves their CLTs for processes with regular paths, and applies these results to parameter estimation in semilinear SPDEs, resolving a prior conjecture.

Findings

Established CLTs for Δ-power variations of stochastic processes.

Derived explicit bias formulas for estimators in SPDEs.

Provided convergence rates and numerical illustrations.

Abstract

Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are proved by using the notion of $\Delta$-power variations, introduced herein, along with the H\"older-Zygmund norms. Consequently, we prove a new central limit theorem for $\Delta$-power variations of the iterated integrals of a fractional Brownian motion (fBm). These abstract results, besides being of independent interest, in the second part of the paper are applied to estimation of the drift and volatility coefficients of semilinear stochastic partial differential equations in dimension one, driven by an additive Gaussian noise white in time and possibly colored in space. In particular, we solve the earlier conjecture from Cialenco, Kim, Lototsky (2019) about existence of a nontrivial bias in the estimators derived by naive approximations of derivatives by finite differences. We give an explicit formula for the bias and derive the convergence rates of the corresponding estimators. Theoretical results are illustrated by numerical examples.