SEM modeling with singular moment matrices Part II: ML-Estimation of sampled stochastic differential equations

Linear stochastic differential equations (SDE) are expressed as an exact discrete model (EDM) and estimated with structural equation models (SEM) and the Kalman filter (KF) algorithm. The SEM likelihood is well defined even for the times series case and the SEM and KF approach yield the same likelihood. The oversampling approach is introduced in order to formulate the EDM on a time grid which is finer than the sampling intervals. This leads to a simple computation of the nonlinear parameter functionals of the EDM. For small discretization intervals, the functionals can be linearized and software permitting only linear parameter restrictions can be used. However, in this case the SEM approach must handle large matrices leading to degraded performance and possible numerical problems. The methods are compared using coupled linear random oscillators with time varying parameters and irregular sampling times.