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

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  • Janák, Josef
  • Reiß, Markus

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.

Suggested Citation

  • Janák, Josef & Reiß, Markus, 2024. "Parameter estimation for the stochastic heat equation with multiplicative noise from local measurements," Stochastic Processes and their Applications, Elsevier, vol. 175(C).
    • Handle: RePEc:eee:spapps:v:175:y:2024:i:c:s0304414924000917
    DOI: 10.1016/j.spa.2024.104385
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    References listed on IDEAS

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    1. Igor Cialenco & Sergey Lototsky, 2009. "Parameter estimation in diagonalizable bilinear stochastic parabolic equations," Statistical Inference for Stochastic Processes, Springer, vol. 12(3), pages 203-219, October.
    2. Cheng, Ziteng & Cialenco, Igor & Gong, Ruoting, 2020. "Bayesian estimations for diagonalizable bilinear SPDEs," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 845-877.
    3. Cialenco, Igor & Glatt-Holtz, Nathan, 2011. "Parameter estimation for the stochastically perturbed Navier-Stokes equations," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 701-724, April.
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