  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 a 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.
