The roughness exponent and its model-free estimation

Understand

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 it works strictly trajectory-wise and requires no probabilistic assumptions.