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](https://en.wikipedia.org/wiki/Zakai_equation), a linear stochastic PDE. The normalized measure satisfies the [Kushner-FKK equation](https://en.wikipedia.org/wiki/Kushner_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?



Why would one work with Kushner-FKK equation over Zakai equation?
The Kushner equation is not suitable for numerical solution, because of its nonlinearity, but it does give the quantity, a normalized measure, you ultimately want. The Zakai equation, in contrast, can be readily solved numerically (Galerkin method), and if it has a unique solution it gives the solution of the Kushner equation upon normalization. So the remaining issue is to prove under which conditions the solution of the Zakai equation is unique. This has been investigated in [The Zakai equation of nonlinear filtering for jump-diffusion observation: existence and uniqueness](https://arxiv.org/abs/1210.4279) (2012).


