2021/11/19 by Xiyue Han, Han, Xiyue, Alexander Schied +1 · 1 citation
Economics, Econometrics and Finance · #26A30 #60F15 #60G22 #60G46 #62G05 #Complex Systems and Time Series Analysis #FOS: Economics and business #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Statistical Finance (q-fin.ST) #Statistics Theory (math.ST) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2111.10301
openalex publication_date 2021/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by pathwise stochastic calculus, we say that a continuous real-valued function x admits the roughness exponent R if the pth 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 Rn(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. 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 Rn)n∈\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 pth variation of x is defined over a sequence of unequally spaced partitions. Our results are illustrated by means of high-frequency financial time series.