Approximation of nonlinear filters for Markov systems with delayed observations

We obtain some approximation results for a class of nonlinear filtering problems with delay in the observation, i.e. systems (X, Y), which can be represented by means of a Markov system (X, Ycirc), in the sense that Y_t = Ycirc_a(t) To this aim we give some general upper bounds which are computed explicitly in the particular case of Markov jump processes with counting observations