The estimation problem starts with an ODE and a set of observations. Two main approaches are currently used to solve the estimation problem.
The embedding method imposes boundary conditions on the ODE and these introduce m extra parameters which must be adjusted as part of the estimation procedure. The resulting boundary value problem must be solved and this could result in additional overhead. The simultaneous method poses the estimation problem as a constrained optimisation problem where the equality constraints are obtained by discretizing the differential equation. This approach could be more efficient in general as the solution of the ODE is not obtained until the process has converged in contrast to the embedding method. However, this means that questions such as error control in the ODE solution become more obscure.
Questions of interest include the equivalence of the two approaches (formally they look rather different), and convergence rates for algorithms. There is particular interest in large sample (large n) properties. The equivalence question, at the level of the necessary conditions, suggests a possible new algorithm which looks like a bit like hybrid scheme.