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For the stochastic heat equation with multiplicative noise we consider theproblem of estimating the diffusivity parameter in front of the Laplaceoperator. Based on local observations in space, we first study an estimatorthat was derived for additive noise. A stable central limit theorem shows thatthis estimator is consistent and asymptotically mixed normal. By taking intoaccount the quadratic variation, we propose two new estimators. Their limitingdistributions exhibit a smaller (conditional) variance and the last estimatoralso works for vanishing noise levels. The proofs are based on localapproximation results to overcome the intricate nonlinearities and on a stablecentral limit theorem for stochastic integrals with respect to cylindricalBrownian motion. Simulation results illustrate the theoretical findings.

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