The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

Abstract

We extend the Kalman-Bucy filter to the case where both the system and observation processes are driven by finite dimensional Lévy 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.

On $$L_p$$-solvability of stochastic integro-differential equations

“…Nonlinear filtering problems and the related equations describing the conditional distributions have been extensively studied in the literature. For results in the case of jump-diffusion models see, for example, [2], [4], [12] and [16].…”