# The Hurst roughness exponent and its model-free estimation

We say that a continuous real-valued function x admits the Hurst roughness exponent H if the p^th variation of x converges to zero if p>1/H and to infinity if p<1/H. For the sample paths of many stochastic processes, such as fractional Brownian motion, the Hurst roughness exponent exists and equals the standard Hurst parameter. In our main result, we provide a mild condition on the Faber–Schauder coefficients of x under which the Hurst roughness exponent exists and is given as the limit of the classical Gladyshev estimates H_n(x). This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because no assumption whatsoever is made on the possible dynamics of the function x. Nonetheless, our proof is probabilistic and relies on a martingale that is hidden in the Faber–Schauder expansion of x. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators that are derived from the sequence (H_n)_n∈ℕ. We also discuss how a dynamic change in the Hurst roughness parameter of a time series can be detected. Finally, we extend our results to the case in which the p^th variation of x is defined over a sequence of unequally spaced partitions. Our results are illustrated by means of high-frequency financial time series.

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