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.

中文翻译：

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Zakai方程的基于核的搭配方法

我们研究了基于核的插值在Zakai方程数值解的非线性滤波中的应用，并试图证明其严格的收敛性。为此，我们明确找到了内核的类别和并置点的结构，在这些条件下迭代插值过程是稳定的。该结果与误差估计中的标准参数一起表明，近似误差受时间步的平方根和来自单步插值的误差的限制。我们的定理与数值实验的结果非常吻合。