Particle filtering with dependent noise

The theory and applications of the particle filter (PF) have developed tremendously during the past two decades. However, there appear to be no version of the PF readily applicable to the case of dependent process and measurement noise. This is in contrast to the Kalman filter, where the case of correlated noise is a standard modification. Further, the fact that sampling continuous time models give dependent noise processes is an often neglected fact in literature. We derive the optimal proposal distribution in the PF for general and Gaussian noise processes, respectively. The main result is a modified prediction step. It is demonstrated that the original Bootstrap particle filter gets a particular simple and explicit form for dependent Gaussian noise. Finally, the practical importance of dependent noise is motivated in terms of sampling of continuous time models.