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
