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Volatility estimation for stochastic PDEs using high-frequency observations

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  • Bibinger, Markus
  • Trabs, Mathias

Abstract

We study the parameter estimation for parabolic, linear, second-order, stochastic partial differential equations (SPDEs) observing a mild solution on a discrete grid in time and space. A high-frequency regime is considered where the mesh of the grid in the time variable goes to zero. Focusing on volatility estimation, we provide an explicit and easy to implement method of moments estimator based on squared increments. The estimator is consistent and admits a central limit theorem. This is established moreover for the joint estimation of the integrated volatility and parameters in the differential operator in a semi-parametric framework. Starting from a representation of the solution of the SPDE with Dirichlet boundary conditions as an infinite factor model and exploiting mixing-type properties of time series, the theory considerably differs from the statistics for semi-martingales literature. The performance of the method is illustrated in a simulation study.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:3005-3052
    DOI: 10.1016/j.spa.2019.09.002
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    References listed on IDEAS

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    Cited by:

    1. Benth, Fred Espen & Schroers, Dennis & Veraart, Almut E.D., 2022. "A weak law of large numbers for realised covariation in a Hilbert space setting," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 241-268.
    2. Hildebrandt, Florian & Trabs, Mathias, 2023. "Nonparametric calibration for stochastic reaction–diffusion equations based on discrete observations," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 171-217.
    3. Patrick Bossert, 2024. "Parameter estimation for second-order SPDEs in multiple space dimensions," Statistical Inference for Stochastic Processes, Springer, vol. 27(3), pages 485-583, October.
    4. Dennis Schroers, 2024. "Robust Functional Data Analysis for Stochastic Evolution Equations in Infinite Dimensions," Papers 2401.16286, arXiv.org, revised Jun 2024.
    5. 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.

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