In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.

在本文中，我们考虑一种非线性滤波模型，其观测值由相关维纳过程和点过程驱动。我们首先推导出一个 Zakai 方程，其解是滤波器解的非归一化概率密度函数。然后，我们应用分裂技术将Zakai方程分解为三个随机微分方程，在此基础上构造分裂近似解并证明其半阶收敛性。此外，我们应用有限差分方法构造了分裂系统的时间半离散近似解，并证明了其对Zakai方程精确解的半阶收敛性。最后，我们提出了一些数值实验来证明理论分析。