SDE - A program package for the simulation, optimal filtering and maximum likelihood estimation of nonlinear Stochastic Differential Equations

Continuous time models with sampled data possess several advantages over conventional time series and panel models (special issue 62:1, 2008, of Statistica Neerlandica). For example, data with unequal time intervals can be treated efficiently, since the dynamic model parameters of the system model are not affected by the measurement process. In the linear case, the nonlinear parameter restrictions of the sampled model can be implemented with specialized Kalman filter software (e.g. LSDE) or with structural equations models (SEM) allowing such nonlinear parameter restrictions. In the nonlinear case, most filtering algorithms are formulated in discrete time, but mixed continuous-discrete approaches are also scattered in the literature. The Mathematica program SDE is a collection of algorithms with consequent focus on the mixed continuous-discrete case: continuous time updates combined with discrete time measurement updates. Included are the classical methods of extended Kalman filtering and higher order nonlinear filters, but also new developments such as the unscented Kalman filter (UKF) and the Gauss-Hermite filter (GHF) using approximations of the filter density. We also use the Edgeworth-Hermite expansion of probability densities to obtain generalized Gauss filters (GGHF) utilizing higher order moments such as skewness and kurtosis.