The Kalman-Bucy Filter for Integrable Levy Processes with Infinite Second Moment

We extend the Kalman-Bucy filter to the case where both the system and observation processes are driven by finite dimensional L´evy processes, but whereas the process driving the system dynamics is square-integrable, that driving the observations is not; however it remains integrable. The main result is that the components of the observation nose that have infinite variance make no contribution to the filtering equations. The key technique used is approximation by processes having bounded jumps.