A novel algorithm to solve the robust DMZ equation in real time

In this note, we extend the algorithm developed in [13] to the most general settings of nonlinear filtering. We rigorously show that under very mild conditions (which essentially say that the growth of the observation |h| is greater than the growth of the drift |f|), the unique nonnegative weak solution of the robust Duncan-Mortensen-Zakai (DMZ) equation can be approximated by solving the same Kolmogorov forward equation (KFE) restricted on a large ball B_R with 0-Dirichlet boundary condition on every time steps (off-line computations) and updating the initial data from the observations (on-line computations). The precise error estimates are obtained to validate the algorithm theoretically. Furthermore, we use 1D cubic sensor as an example to show the efficiency of our algorithm, where the Hermite spectral method (HSM) is adopted to compute the off-line data. The detailed formulation of HSM is illustrated.