Deterministic and stochastic Gaussian particle smoothing

In this article we study inference problems in non-linear dynamical systems. In particular we are concerned with assumed density approaches to filtering and smoothing. In models with uncorrelated (but dependent) state and observation, the extended Kalman filter and the unscented Kalman filter break down. We show that the Gaussian particle filter and the one-step unscented Kalman filter make less assumptions and potentially form useful filters for this class of models. We construct a symmetric smoothing pass for both filters that does not require the dynamics to be invertible. We investigate the characteristics of the methods in an interesting problem from mathematical finance. Among others we find that smoothing helps, in particular for the deterministic one-step unscented Kalman filter.