Hypothesis testing for stochastic PDEs driven by additive noise

TL;DR

This paper develops optimal hypothesis tests for the drift coefficient in stochastic fractional heat equations driven by additive noise, using likelihood-based statistics and asymptotic analysis.

Contribution

It introduces the concept of asymptotically most powerful tests and derives explicit forms in large time and Fourier mode regimes.

Findings

Explicit forms of the most powerful tests for different asymptotic regimes

Derivation of the cumulant generating function of the log-likelihood ratio

Large deviation results for increasing time and Fourier modes

Abstract

We study the simple hypothesis testing problem for the drift coefficient for stochastic fractional heat equation driven by additive noise. We introduce the notion of asymptotically the most powerful test, and find explicit forms of such tests in two asymptotic regimes: large time asymptotics, and increasing number of Fourier modes. The proposed statistics are derived based on Maximum Likelihood Ratio. Additionally, we obtain a series of important technical results of independent interest: we find the cumulant generating function of the log-likelihood ratio; obtain sharp large deviation type results for $T\to\infty$ and $N\to\infty$.