The Kalman-Bucy Filter for Integrable L\'{e}vy Processes With Infinite Second Moment

TL;DR

This paper extends the Kalman-Bucy filter to systems driven by Lévy processes with infinite second moments, using approximation techniques to handle unbounded jumps, thus broadening its applicability.

Contribution

It introduces a novel extension of the Kalman-Bucy filter for Lévy processes with infinite second moments, employing approximation methods for unbounded jumps.

Findings

Successfully extends filtering to Lévy processes with infinite second moments

Demonstrates approximation by bounded jump processes

Broadens the applicability of Kalman-Bucy filtering techniques

Abstract

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.