Backward and forward filtering under the weak Hörmander condition

We derive the forward and backward filtering equations for a class of degenerate partially observable diffusions, satisfying the weak Hörmander condition. Our approach is based on the Hölder theory for degenerate SPDEs that allows to pursue the direct approaches proposed by Krylov and Zatezalo, and Veretennikov, avoiding the use of general results from filtering theory. As a by-product we also provide existence, regularity and estimates for the filtering density.