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Statistical inference for SPDEs: an overview

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  • Igor Cialenco

    (Illinois Institute of Technology)

Abstract

The aim of this work is to give an overview of the recent developments in the area of statistical inference for parabolic stochastic partial differential equations. Significant part of the paper is devoted to the spectral approach, which is the most studied sampling scheme under which the observations are done in the Fourier space over some finite time interval. We also discuss the practically important case of discrete sampling of the solution. Other relevant methodologies and some open problems are briefly addressed over the course of the manuscript.

Suggested Citation

  • Igor Cialenco, 2018. "Statistical inference for SPDEs: an overview," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 309-329, July.
    • Handle: RePEc:spr:sistpr:v:21:y:2018:i:2:d:10.1007_s11203-018-9177-9
    DOI: 10.1007/s11203-018-9177-9
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    References listed on IDEAS

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    1. M. Huebner, 1999. "Asymptotic Properties of the Maximum Likelihood Estimator for Stochastic PDEs Disturbed by Small Noise," Statistical Inference for Stochastic Processes, Springer, vol. 2(1), pages 57-68, January.
    2. S. Lototsky, 2003. "Parameter Estimation for Stochastic Parabolic Equations: Asymptotic Properties of a Two-Dimensional Projection-Based Estimator," Statistical Inference for Stochastic Processes, Springer, vol. 6(1), pages 65-87, January.
    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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    5. Lototsky, Sergey V. & Rosovskii, Boris L., 1999. "Spectral asymptotics of some functionals arising in statistical inference for SPDEs," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 69-94, January.
    6. Loges, Wilfried, 1984. "Girsanov's theorem in Hilbert space and an application to the statistics of Hilbert space- valued stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 17(2), pages 243-263, July.
    7. M.L. Kleptsyna & A. Le Breton & M.-C. Roubaud, 2000. "Parameter Estimation and Optimal Filtering for Fractional Type Stochastic Systems," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 173-182, January.
    8. Goldys, B. & Maslowski, B., 2002. "Parameter Estimation for Controlled Semilinear Stochastic Systems: Identifiability and Consistency," Journal of Multivariate Analysis, Elsevier, vol. 80(2), pages 322-343, February.
    9. Cialenco, Igor & Xu, Liaosha, 2015. "Hypothesis testing for stochastic PDEs driven by additive noise," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 819-866.
    10. Koski, Timo & Loges, Wilfried, 1985. "Asymptotic statistical inference for a stochastic heat flow problem," Statistics & Probability Letters, Elsevier, vol. 3(4), pages 185-189, July.
    11. M. Huebner & S. Lototsky, 2000. "Asymptotic Analysis of the Sieve Estimator for a Class of Parabolic SPDEs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(2), pages 353-370, June.
    12. 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.
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    Cited by:

    1. 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.
    2. Dennis Schroers, 2024. "Robust Functional Data Analysis for Stochastic Evolution Equations in Infinite Dimensions," Papers 2401.16286, arXiv.org, revised Jun 2024.
    3. Pavel Kříž & Leszek Szała, 2020. "The Combined Estimator for Stochastic Equations on Graphs with Fractional Noise," Mathematics, MDPI, vol. 8(10), pages 1-21, October.
    4. 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.
    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.
    6. 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.
    7. 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.
    8. di Nunno, Giulia & Ortiz–Latorre, Salvador & Petersson, Andreas, 2023. "SPDE bridges with observation noise and their spatial approximation," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 170-207.
    9. 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.

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