Efficiency of an approximate filter for a particular class of nonlinear diffusions with observations corrupted by small noise

The asymptotic behaviour of a nonlinear continuous time approximate filter when the variance of the observation noise tends to 0 is investigated. We consider a particular class of signals modeled by a two-dimensional quasi-linear diffusion from which only one of the components is noisy, and we assume that a one-dimensional linear function of the signal, depending only of the unnoisy component, is observed in a low noise channel. Under some detectability assumptions the unobserved signal can be restored by means of an approximate nonlinear filter. We establish that the filtering error converges to 0 and we give an upper bound for the convergence rate. The efficiency of the approximate filter is compared with the efficiency of the optimal filter and the order of magnitude of the error between the two filters, as the observation noise vanishes, is obtained. A more general case is briefly presented