Splitting-up discretization for Kushner´s equation of nonlinear filtering

The great majority of numerical schemes for optimal nonlinear filtering of randomly perturbed dynamical systems deal with the Zakai equation. Unfortunately, in spite of its popularity, the Zakai equation has serious deficiencies as a computational tool, including fast dissipation of the solution and an intermittency effect which manifests itself in the appearance of rare but very large peaks. It appears that Kushner´s equation of nonlinear filtering (1977) is not subject to these problems. In this paper we present (without proofs) two operator splitting-up approximations for the Kushner equation. It is shown that one of them is equivalent to the splitting-up approximation to the Zakai equation with normalization on each step. We also discuss the rate of convergence of these schemes