Robust Approximation in a Filtering Problem with Real State Space and Counting Observations

Let (Xt,Yt) be a pure jump Markov process, where Xt takes values in bf R and Yt 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