Explicit solution of DMZ equation

We consider the explicit solution of the Duncan-Mortensen-Zakai (DMZ) equation for the finite dimensional filtering system. We show that the nonlinear filtering system under the Hu-Yau conditions can be solved explicitly with an arbitrary initial condition by solving and system of ordinary differential equations and a Kolmogorov-type equation. Let n be the dimension of state space. We show that we need only n sufficient statistics in order to solve the DMZ equation.