Explicit solution of DMZ equation in nonlinear filtering via solution of ODEs

In this note, we develop a real-time and accurate solution for nonlinear filtering problems based on the Gaussian distribution. Specifically, we present an explicit solution of the Duncan-Mortensen-Zakai (DMZ) equation of the Yau filtering system, which includes the linear filtering system and exact filtering system. The solution is given in terms of a solution of a system of ordinary differential equations. In particular, our method can be implemented in hardware. The complexity of our algorithms is the same as those of Kalman-Bucy filters in the case of linear filtering systems.