Feedback Particle Filter with Correlated Noises

Motivated by the mean-field game theory, the feedback particle filter (FPF) for the signal-observation nonlinear filtering (NLF) model with independent white noises, has been developed in [23] for the first time. In this paper, we shall extend this algorithm to the case where the scalar signal process is correlated with the scalar observation process. The equation that the control inputs (K, u) satisfied has been derived by minimizing the Kullback-Leibler (K-L) divergence of the conditional density and the conditional posterior empirical distribution of the controlled particles. Then we show rigorously that the control inputs obtained is consistent, in the sense that if the initial conditional density and the empirical distribution are the same, so are the posterior ones. The explicit expression for the control input u is given if K is obtained. The numerical simulation of a scalar NLF problem with transition phenomenon has been solved by our algorithm with satisfactory performance not only in accuracy but also in efficiency.