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Backward and forward stochastic partial differential equations associated with a non linear filtering problem

We associate to a non linear filtering problem a pair of stochastic PDE's. They play, in the filtering set up, the role of the backward and forward Kolmogorov equations for unconditionned diffusions, We treat separately the two cases where the observation noise is independent of, and correlated with the signal process. In the first case, we obtain directly a "robust" form of the associated equations.