Approximate filters for a nonlinear discrete time filtering problem with small observation noise

The asymptotic behavior is studied of a nonlinear one-dimensional filtering discrete time problem, as some parameter ∈ tends to 0. The case is treated of a nonlinear discrete time problem coming from a continuous time one with small observation noise, the step time discretization being equal to the intensity of this noise. A finite dimensional approximate filter is proposed, and results concerning estimations of its performance are stated and proved. The proof of the result concerning the error between the approximate filter and the optimal one makes use of probability changes and differentiation with respect to the initial condition. Then the main ideas of how to treat other cases are given