Kernel-based collocation methods for Zakai equations
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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.
KeywordsZakai equations Kernel-based interpolation Stochastic partial differential equations Radial basis functions
Mathematics Subject Classification60H15 65M70 93E11
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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