Marchetti, CharlesWe obtain an asymptotic expansion of the conditional distribution for a homogeneous Markov chain in the filtering problem. In a first case, we consider a regular perturbation situation, and in a second case, a singular perturbation situation, according the time scale of the scale of the observe process. In all cases, we obtain an asymptotic expansion which the terms are calculated easier by decentralization and aggregation.en-USTechnical Report