TY - JOUR
T1 - A Splitting Method for Nonlinear Filtering Problems with Diffusive and Point Process Observations
AU - Zhang , Fengshan
AU - Zou , Yongkui
AU - Chai , Shimin
AU - Cao , Yanzhao
JO - Communications in Computational Physics
VL - 4
SP - 996
EP - 1020
PY - 2024
DA - 2024/10
SN - 36
DO - http://doi.org/10.4208/cicp.OA-2024-0075
UR - https://global-sci.org/intro/article_detail/cicp/23484.html
KW - Nonlinear filtering problem, Zakai equation, splitting-up technique, error analysis.
AB - <p style="text-align: justify;">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.</p>