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Parameter estimation for a linear parabolic SPDE model in two space dimensions with a small noise

Author

Listed:
  • Yozo Tonaki

    (Osaka University)

  • Yusuke Kaino

    (Kobe University)

  • Masayuki Uchida

    (Osaka University
    Osaka University and JST CREST)

Abstract

We study parameter estimation for a linear parabolic second-order stochastic partial differential equation (SPDE) in two space dimensions with a small dispersion parameter using high frequency data with respect to time and space. We set two types of Q-Wiener processes as a driving noise. We provide minimum contrast estimators of the coefficient parameters of the SPDE appearing in the eigenfunctions of the differential operator of the SPDE based on the thinned data in space, and approximate the coordinate process based on the thinned data in time. Moreover, we propose an estimator of the drift parameter using the fact that the coordinate process is the Ornstein-Uhlenbeck process and statistical inference for diffusion processes with a small noise. We also give an example and simulation results for the proposed estimators.

Suggested Citation

  • Yozo Tonaki & Yusuke Kaino & Masayuki Uchida, 2024. "Parameter estimation for a linear parabolic SPDE model in two space dimensions with a small noise," Statistical Inference for Stochastic Processes, Springer, vol. 27(1), pages 123-179, April.
    • Handle: RePEc:spr:sistpr:v:27:y:2024:i:1:d:10.1007_s11203-023-09301-2
    DOI: 10.1007/s11203-023-09301-2
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    References listed on IDEAS

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    1. Yusuke Kaino & Masayuki Uchida, 2018. "Hybrid estimators for small diffusion processes based on reduced data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(7), pages 745-773, October.
    2. Igor Cialenco, 2018. "Statistical inference for SPDEs: an overview," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 309-329, July.
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    6. Tetsuya Kawai & Masayuki Uchida, 2023. "Adaptive inference for small diffusion processes based on sampled data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(6), pages 643-696, August.
    7. Hildebrandt, Florian, 2020. "On generating fully discrete samples of the stochastic heat equation on an interval," Statistics & Probability Letters, Elsevier, vol. 162(C).
    8. Yozo Tonaki & Yusuke Kaino & Masayuki Uchida, 2023. "Parameter estimation for linear parabolic SPDEs in two space dimensions based on high frequency data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 50(4), pages 1568-1589, December.
    9. 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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    11. Masayuki Uchida, 2004. "Estimation for Discretely Observed Small Diffusions Based on Approximate Martingale Estimating Functions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(4), pages 553-566, December.
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