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SPDE bridges with observation noise and their spatial approximation

Author

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  • di Nunno, Giulia
  • Ortiz–Latorre, Salvador
  • Petersson, Andreas

Abstract

This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:158:y:2023:i:c:p:170-207
    DOI: 10.1016/j.spa.2023.01.007
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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. Goldys, B. & Maslowski, B., 2008. "The Ornstein-Uhlenbeck bridge and applications to Markov semigroups," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1738-1767, October.
    3. Cui, Jianbo & Hong, Jialin & Sun, Liying, 2021. "Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 55-93.
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