Gaussian filtering and smoothing for continuous-discrete dynamic systems

This paper is concerned with Bayesian optimal filtering and smoothing of non-linear continuous-discrete state space models, where the state dynamics are modeled with non-linear Itô-type stochastic differential equations, and measurements are obtained at discrete time instants from a non-linear measurement model with Gaussian noise. We first show how the recently developed sigma-point approximations as well as the multi-dimensional Gauss–Hermite quadrature and cubature approximations can be applied to classical continuous-discrete Gaussian filtering. We then derive two types of new Gaussian approximation based smoothers for continuous-discrete models and apply the numerical methods to the smoothers. We also show how the latter smoother can be efficiently implemented by including one additional cross-covariance differential equation to the filter prediction step. The performance of the methods is tested in a simulated application.

► Sigma-point approximations of non-linear continuous-discrete Gaussian filters.
► New Gaussian smoothers for non-linear continuous-discrete state space models.
► The first smoother via Gaussian approximation to the formal smoothing equations.
► The second smoother via continuous-time limit of the discrete-time smoother.
► The third smoother via formal manipulation of the second to an efficient form.