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

## TL;DR

This paper develops and compares new estimators for the diffusivity parameter in the stochastic heat equation with multiplicative noise, demonstrating improved statistical properties and robustness through theoretical analysis and simulations.

## Contribution

It introduces two novel estimators that account for quadratic variation, offering smaller variance and applicability at low noise levels, with proven asymptotic properties.

## Key findings

- New estimators have smaller (conditional) variance.
- Estimates remain consistent and asymptotically normal.
- Simulation results confirm theoretical advantages.

## 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 that was derived 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.