  We study the problem of estimating parameters of the limiting equation of a
multiscale diffusion in the case of averaging and homogenization, given data
from the corresponding multiscale system. First, we review some recent results
that make use of the maximum likelihood of the limiting equation. In
particular, it has been shown that in the averaging case, the MLE will be
asymptotically consistent in the limit while in the homogenization case, the
MLE will be asymptotically consistent only if we subsample the data. Then, we
focus on the problem of estimating the diffusion coefficient. We suggest a
novel approach that makes use of the total $p$-variation, as defined in the
theory of rough paths and avoids the subsampling step. The method is applied to
a multiscale OU process.
