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Kernel-based collocation methods for Zakai equations

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A Correction to this article was published on 20 November 2019

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Abstract

We examine an application of the kernel-based interpolation to numerical solutions for Zakai equations in nonlinear filtering, and aim to prove its rigorous convergence. To this end, we find the class of kernels and the structure of collocation points explicitly under which the process of iterative interpolation is stable. This result together with standard argument in error estimation shows that the approximation error is bounded by the order of the square root of the time step and the error that comes from a single step interpolation. Our theorem is well consistent with the results of numerical experiments.

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Change history

  • 20 November 2019

    The claim of Lemma 3.7 in [1] needs to be modified. This lemma states that the sum of the absolute value of the cardinal functions is bounded with respect to the approximation parameters and the spatial variables.

  • 20 November 2019

    The claim of Lemma 3.7 in [1] needs to be modified. This lemma states that the sum of the absolute value of the cardinal functions is bounded with respect to the approximation parameters and the spatial variables.

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Acknowledgements

The author is thankful to the anonymous reviewers for their careful reading of previous versions of this paper and their constructive comments. This study is partially supported by JSPS KAKENHI Grant No. JP17K05359.

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  1. Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, W8-28, 2-12-1, Ookayama, Meguro-ku, Tokyo, 152-8550, Japan

    Yumiharu Nakano

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  1. Yumiharu Nakano
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Correspondence to Yumiharu Nakano.

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Nakano, Y. Kernel-based collocation methods for Zakai equations. Stoch PDE: Anal Comp 7, 476–494 (2019). https://doi.org/10.1007/s40072-019-00132-y

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  • DOI: https://doi.org/10.1007/s40072-019-00132-y

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