In this report, a new interpretation of the main part of our study on asymptotic behaviour of reduced-order filters in [6] is given. Approximate filters are proposed for semi-linear and nonlinear stochastic systems with colored noises. Basically these filters are defined as those which are optimal when the noises are white. Their long time behavior is investigate- d and their asymptotic optimality is shown in two cases under some reasonable assumptions. At first the case of a system where the signal and observation dynamics are linear with respect to the state is considered; the approximate filter is a Kalman filter and the asymptotic analysis relies on a representati- on of the optimal filter which extends a formula known for a linear system with white noises and non-Gaussian initial condition. Then the case of a nonlinear system with limiting ergodic behavior is analyzed; here the approximate filter appears as the optimal filter corresponding to an incorrect prior distribution and the asymptotic study uses the results of Ocone and Pardoux on the asymptotic stability of optimal filters with respect to their initial condition
