The roughness exponent and its model-free estimation

Han, Xiyue
Schied, Alexander

Publication date

August 2023

Language

English

Abstract

Motivated by pathwise stochastic calculus, we say that a continuous real-valued function $x$ admits the roughness exponent $R$ if the $p^{\text{th}}$ variation of $x$ converges to zero if $p>1/R$ and to infinity if $p<1/R$. For the sample paths of many stochastic processes, such as fractional Brownian motion, the 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 roughness exponent exists and is given as the limit of the classical Gladyshev estimates $\widehat R_n(x)$. This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because no assumption whatsoever...

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The roughness exponent and its model-free estimation

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The roughness exponent and its model-free estimation

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