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 Ito-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.