An averaging principle for filtering a jump process with pointprocess observations

A proof of the following result is given. Le X_t and Y_t be two jump processes which modulate the intensity of a multivariate point process N _t, and suppose that the process X_t is a fast' Markov chain with a unique invariant probability distribution. Then the filtering equations for Y_t can be obtained by considering, instead of the original problem, the averaged problem where the intensity is replaced by the averaged intensity