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We introduce the concept of finite $\gamma$-scaled quadratic variation alonga sequence of partitions for paths on a given interval. This concept, withhistorical roots in the study of Gaussian processes by Gladyshev (1961) andKlein \& Gin\'e (1975), includes the fractional Brownian motion (fBM) withHurst index $H$, which has finite $1-2H$-scaled quadratic variation. We showthat a path that is controlled by a path with finite $\gamma$-scaled quadraticvariation in the sense of M. Gubinelli inherits this property, and thecorresponding scaled quadratic variation satisfies an It\^o-isometry typeformula. Moreover, we prove quantitative error bounds that establish arelationship between the convergence rates of the scaled quadratic variation ofthe controlled path and that of the controlling path. Additionally, weintroduce a consistent estimator for the parameter $\gamma$ based on a singlesample path, complete with quantitative error bounds. We apply these results tothe parameter estimation for fractional diffusions. Our findings specifyconvergence rates for the estimation of both the Hurst index and the parametersin the noise vector fields. The paper concludes with numerical experiments thatsubstantiate our theoretical findings.

Scaled quadratic variation for controlled rough paths and parameter estimation of fractional diffusions