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 a nd Y-t is a counting process. We compare the filter of this system and a fi lter of a suitably modified system. We compute an explicit bound for the di stance in the so-called bounded Lipschitz metric between the two filters. F inally we show how to use this bound to construct a discrete space approxim ation of the filter.