Parameter estimation for the stochastic heat equation with multiplicative noise from local measurements

Josef Janák and Markus Reiß

Stochastic Processes and their Applications, 2024, vol. 175, issue C

Abstract: For the stochastic heat equation with multiplicative noise we consider the problem of estimating the diffusivity parameter in front of the Laplace operator. Based on local observations in space, we first study an estimator, derived in Altmeyer and Reiß (2021) for additive noise. A stable central limit theorem shows that this estimator is consistent and asymptotically mixed normal. By taking into account the quadratic variation, we propose two new estimators. Their limiting distributions exhibit a smaller (conditional) variance and the last estimator also works for vanishing noise levels. The proofs are based on local approximation results to overcome the intricate nonlinearities and on a stable central limit theorem for stochastic integrals with respect to cylindrical Brownian motion. Simulation results illustrate the theoretical findings.

Keywords: Local measurements; Stochastic partial differential equation; Multiplicative noise; Drift estimation; Augmented MLE; Martingale representation theorem; Stable limit theorem (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spa.2024.104385

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