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 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 $(\widehat R_n)_{n\in\mathbb N}$. We also discuss how a dynamic change in the roughness parameter of a time series can be detected. Finally, we extend our results to the case in which the $p^{\text{th}}$ variation of $x$ is defined over a sequence of unequally spaced partitions. Our results are illustrated by means of high-frequency financial time series.

翻译：受路径随机微积分的启发，我们称一个连续实值函数$x$具有粗糙度指数$R$，若其$p$次变差在$p>1/R$时收敛于零，在$p<1/R$时收敛于无穷大。对于许多随机过程（如分数布朗运动）的样本轨道，粗糙度指数存在且等于标准赫斯特参数。在我们的主要结果中，我们给出了$x$的Faber–Schauder系数的一个温和条件，在此条件下粗糙度指数存在且等于经典Gladyshev估计$\widehat R_n(x)$的极限。这一结果可视为Gladyshev估计在完全无模型设定下的强相合性结果，因为对函数$x$可能的动态过程未作任何假设。尽管如此，我们的证明是概率性的，并依赖于隐藏在$x$的Faber–Schauder展开中的鞅。由于Gladyshev估计不具有尺度不变性，我们构造了几个基于序列$(\widehat R_n)_{n\in\mathbb N}$导出的尺度不变估计量。我们还讨论了如何检测时间序列中粗糙度参数的动态变化。最后，我们将结果推广到$x$的$p$次变差定义在一系列非均匀划分上的情形。我们的结果通过高频金融时间序列进行了验证。