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 solutions 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 (2012).

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 (2012).