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

Abstract: We introduce the concept of finite $\gamma$-scaled quadratic variation along a sequence of partitions for paths on a given interval. This concept, with historical roots in the study of Gaussian processes by Gladyshev (1961) and Klein & Gin\'e (1975), includes the fractional Brownian motion (fBM) with Hurst index $H$, which has finite $1-2H$-scaled quadratic variation. We show that a path that is controlled by a path with finite $\gamma$-scaled quadratic variation in the sense of M. Gubinelli inherits this property, and the corresponding scaled quadratic variation satisfies an It^o-isometry type formula. Moreover, we prove quantitative error bounds that establish a relationship between the convergence rates of the scaled quadratic variation of the controlled path and that of the controlling path. Additionally, we introduce a consistent estimator for the parameter $\gamma$ based on a single sample path, complete with quantitative error bounds. We apply these results to the parameter estimation for fractional diffusions. Our findings specify convergence rates for the estimation of both the Hurst index and the parameters in the noise vector fields. The paper concludes with numerical experiments that substantiate our theoretical findings.