A Splitting Method for Nonlinear Filtering Problems with Diffusive and Point Process Observations (Communications in Computational Physics, 2024)
This paper aims to develop and analyze a comprehensive discretizedsplitting-up numerical scheme for the Zakai equation. This equation arises from anonlinear filtering problem, where observations incorporate noise modeled by pointprocesses and Wiener processes. Initially, we introduce a regularization parameter andemploy a splitting-up approach to break down the Zakai equation into two stochasticdifferential equations and a partial differential equation (PDE). Subsequently, we employ a finite difference scheme for the temporal dimension and the spectral Galerkinmethod for the spatial dimension to achieve full discretization of these equations. Thisresults in a numerical solution for the Zakai equation using the splitting-up technique.We demonstrate that this numerical solution converges to the exact solution with aconvergence order of $\frac{1}{2}.$ Additionally, we conduct several numerical experiments toillustrate and validate our theoretical findings.