In stochastic filtering you are interested in a process called the optimal filter $\pi_t$ which is a probability measure(d stochastic process). You can consider the unnormalized version $V_t$.
The unnormalized measure satisfies the Zakai equation, a linear stochastic PDE. The normalized measure satisfies the Kushner-FKK equation, a nonlinear stochastic PDE.
If you solve the Zakai equation, you can simply normalize to get $\pi_t$.
Is there any reason to work with the Kushner-FKK equation directly? Perhaps some numerics?