  It is well known that the nonlinear filtering problem has important
applications in both military and civil industries. The central problem of
nonlinear filtering is to solve the Duncan-Mortensen-Zakai (DMZ) equation in
real time and in a memoryless manner. In this paper, we shall extend the
algorithm developed previously by S.-T. Yau and the second author to the most
general setting of nonlinear filterings, where the explicit time-dependence is
in the drift term, observation term, and the variance of the noises could be a
matrix of functions of both time and the states. To preserve the off-line
virture of the algorithm, necessary modifications are illustrated clearly.
Moreover, it is shown rigorously that the approximated solution obtained by the
algorithm converges to the real solution in the $L^1$ sense. And the precise
error has been estimated. Finally, the numerical simulation support the
feasibility and efficiency of our algorithm.
