  We study the problem of parameter estimation using maximum likelihood for
fast/slow systems of stochastic differential equations. Our aim is to shed
light on the problem of model/data mismatch at small scales. We consider two
classes of fast/slow problems for which a closed coarse-grained equation for
the slow variables can be rigorously derived, which we refer to as averaging
and homogenization problems. We ask whether, given data from the slow variable
in the fast/slow system, we can correctly estimate parameters in the drift of
the coarse-grained equation for the slow variable, using maximum likelihood. We
show that, whereas the maximum likelihood estimator is asymptotically unbiased
for the averaging problem, for the homogenization problem maximum likelihood
fails unless we subsample the data at an appropriate rate. An explicit formula
for the asymptotic error in the log likelihood function is presented. Our
theory is applied to two simple examples from molecular dynamics.
