Year: 2024
Author: Fengshan Zhang, Yongkui Zou, Shimin Chai, Yanzhao Cao
Communications in Computational Physics, Vol. 36 (2024), Iss. 4 : pp. 996–1020
Abstract
This paper aims to develop and analyze a comprehensive discretized splitting-up numerical scheme for the Zakai equation. This equation arises from a nonlinear filtering problem, where observations incorporate noise modeled by point processes and Wiener processes. Initially, we introduce a regularization parameter and employ a splitting-up approach to break down the Zakai equation into two stochastic differential equations and a partial differential equation (PDE). Subsequently, we employ a finite difference scheme for the temporal dimension and the spectral Galerkin method for the spatial dimension to achieve full discretization of these equations. This results 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 a convergence order of $\frac{1}{2}.$ Additionally, we conduct several numerical experiments to illustrate and validate our theoretical findings.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2024-0075
Communications in Computational Physics, Vol. 36 (2024), Iss. 4 : pp. 996–1020
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 25
Keywords: Nonlinear filtering problem Zakai equation splitting-up technique error analysis.