Nonlinear filtering of smooth signals

A nonlinear online Kalman type filter is proposed for the estimation of unknown function S(t) with the known smoothness)3 for, the diffusion observed process with, small, of the order epsilon(2), diffusion coefficient. Assuming that the drift coefficient of the observed process depends on an unknown function S(t), we propose an approach to the analysis of this estimator based on the Lyapunov's functions method. The best possible rate of convergence of risks to 0, as epsilon -> 0, is proven for beta <= 2.