The Kalman-Bucy Filter for Integrable L\'{e}vy 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\'{e}vy processes, but whereas the process driving the system dynamics is square-integrable, that driving the observations is not; however it remains integrable. The key technique used is approximation by processes having bounded jumps.