A filtering problem with counting observations: approximation with error bounds

Abstract

We consider a pure jump Markov process (Xt Yt ) with discrete state space. We suppose that the state Xt is not observable and that the observation Yt is a counting process. We construct an approximation for the filter of Xt given (Ys s ≤ t), by means of a family of piecewise constant processes, depending on the value of Yt and on the time discretization parameter. Moreover we give an explicit error bound for the convergence of the scheme