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Parameter estimation for discretely sampled stochastic heat equation driven by space-only noise

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  • Cialenco, Igor
  • Kim, Hyun-Jung

Abstract

We derive consistent and asymptotically normal estimators for the drift and volatility parameters of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain. We consider both the full space and the bounded domain. We establish the exact spatial regularity of the solution, which in turn, using power-variation arguments, allows building the desired estimators. We show that naive approximations of the derivatives appearing in the power-variation based estimators may create nontrivial biases, which we compute explicitly. The proofs are rooted in Malliavin–Stein’s method.

Suggested Citation

  • Cialenco, Igor & Kim, Hyun-Jung, 2022. "Parameter estimation for discretely sampled stochastic heat equation driven by space-only noise," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 1-30.
    • Handle: RePEc:eee:spapps:v:143:y:2022:i:c:p:1-30
    DOI: 10.1016/j.spa.2021.09.012
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    References listed on IDEAS

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    1. Igor Cialenco, 2018. "Statistical inference for SPDEs: an overview," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 309-329, July.
    2. Bibinger, Markus & Trabs, Mathias, 2020. "Volatility estimation for stochastic PDEs using high-frequency observations," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3005-3052.
    3. 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.
    4. Igor Cialenco & Hyun-Jung Kim & Sergey V. Lototsky, 2020. "Statistical analysis of some evolution equations driven by space-only noise," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 83-103, April.
    5. 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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