Kernel-based collocation methods for Zakai equations

TL;DR

This paper develops a kernel-based collocation method for solving Zakai equations in nonlinear filtering, providing theoretical convergence guarantees and validating results through numerical experiments.

Contribution

It introduces a new kernel-based interpolation approach with explicit kernel and collocation point structures, ensuring stability and convergence for Zakai equation solutions.

Findings

Error bound proportional to square root of time step

Stable iterative interpolation process established

Numerical experiments confirm theoretical convergence

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.