Finite-dimensional filters with nonlinear drift X: explicit solution of DMZ equation

In this note, we consider the explicit solution of Duncan-Mortensen-Zakai (DMZ) equation for the finite-dimensional filtering system. We show that Yau's (1990, 1994) filtering system (/spl delta/f/sub j///spl delta/x/sub i/)-(/spl delta/f/sub i///spl delta/x/sub i/)=c/sub ij/=constant for all (i,j) can be solved explicitly with an arbitrary initial condition by solving a 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.