A Filtering Problem with Counting Observations: Error Bounds due to the Uncertainty on the Infinitesimal Parameters

Let (X,Y) be a pure jump Markov process with discrete state space. Let the state X be not observable and the observation Y be a counting process. We are interested in the filter of X given Y and in its dependence on the model. More precisely we compare this filter with the filter of another system which differs from the previous one only by the infinitesimal parameters and the initial distribution, and we give an explicit bound for the distance in variation norm between the two filters. Finally we use this bound to examine how much a discrete time approximation procedure is affected by a slight error in the model and, in a special case, to examine the error due to the use of a finite state space model instead of an infinite one.