Asymptotic analysis of the optimal filtering problem for two-dimensional diffusions measured in a low noise channel

The problem of the optimal filtering of a single output two-dimensional diffusion process with nonlinear drift, transmitted through a monotone nonlinear low noise channel is considered. Two types of signals are analyzed, a nonlinear autoregressive (NAR) diffusion process and a nonlinear autoregressive process with moving average (NARMA). A singular perturbation method is used for the construction of an approximate solution to Zakai's equation for the unnormalized probability density function of the signal. Then the leading term in the asymptotic expansion of the minimal mean square estimation error (MMSEE) is found for NAR and NARMA signals. It is shown that NAR signals have the perfect filtering property, that is, the MMSEE vanishes as the noise vanishes. For NARMA processes conditions for perfect and for imperfect filtering are given. These conditions reduce to the minimum phase property in the linear case. First-order approximations to the optimal filters for signals with perfect filtering are given. It is shown that in case of imperfect filtering the measured variable can be perfectly filtered while the MMSEE of the other variable remains O(1) as the noise vanishes. The gains of the two-dimensional approximate filters are explicitly expressible in terms of the coefficients and do not require additional differential equations for their calculation, as does the extended Kalman filter (EKF). Numerical simulations that compare the performance of various filters are presented.