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 filtering system (([[differential] [f.sub.i]/[differential] [x.sub.i]) - ([differential] [f.sub.i]/[differential][x.sub.j]) = [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. Index Terms--Duncan-Mortensen-Zakai (DMZ) equation, finite-dimensional filter, Kolmogorov equation, nonlinear drift.