  We consider the inference problem for parameters in stochastic differential
equation models from discrete time observations (e.g. experimental or
simulation data). Specifically, we study the case where one does not have
access to observations of the model itself, but only to a perturbed version
which converges weakly to the solution of the model. Motivated by this
perturbation argument, we study the convergence of estimation procedures from a
numerical analysis point of view. More precisely, we introduce appropriate
consistency, stability, and convergence concepts and study their connection. It
turns out that standard statistical techniques, such as the maximum likelihood
estimator, are not convergent methodologies in this setting, since they fail to
be stable. Due to this shortcoming, we introduce and analyse a novel inference
procedure for parameters in stochastic differential equation models which turns
out to be convergent. As such, the method is particularly suited for the
estimation of parameters in effective (i.e. coarse-grained) models from
observations of the corresponding multiscale process. We illustrate these
theoretical findings via several numerical examples.
