Robust approximation in a filtering problem with real state space and counting observations

Let (X-t, Y-t) be a pure jump Markov process, where X-t takes values in R and Y-t is a counting process. We compare the filter of this system and a filter of a suitably modified system. We compute an explicit bound for the distance in the so-called bounded Lipschitz metric between the two filters. Finally we show how to use this bound to construct a discrete space approximation of the filter.