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We study the problem of estimating parameters of the limiting equation of amultiscale diffusion in the case of averaging and homogenization, given datafrom the corresponding multiscale system. First, we review some recent resultsthat make use of the maximum likelihood of the limiting equation. Inparticular, it has been shown that in the averaging case, the MLE will beasymptotically consistent in the limit while in the homogenization case, theMLE will be asymptotically consistent only if we subsample the data. Then, wefocus on the problem of estimating the diffusion coefficient. We suggest anovel approach that makes use of the total $p$-variation, as defined in thetheory of rough paths and avoids the subsampling step. The method is applied toa multiscale OU process.

Coarse-grained modeling of multiscale diffusions: the p-variation estimates