Asymptotic study of estimation in filtering for linear systems with jump parameters

In this work, we study the nonlinear filtering problem for a linear diffusion process X, the coefficients of which are fed by a Markovian jump process Z. The state process (Z,X) is assumed to be observed with additive observation noise of order ε. We derive approximate finite dimensional filters which are solutions of stochastic differential equations driven by the observation process; they are asymptotically efficient as ε→0. Upper bounds for the corresponding errors are given