THE ROUGHNESS EXPONENT AND ITS MODEL-FREE ESTIMATION

• 成果类型: Article
• 署名作者: Han, Xiyue; Schied, Alexander
• 署名单位: University of Waterloo
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2135
• 发表日期: 2025
• 页码: 1049-1082
• 关键词: quadratic variation; pth variation; SPACES

摘要

Motivated by pathwise stochastic calculus, we say that continuous real-valued function x admits the roughness exponent R if the pth variation of x converges to zero for p > 1/R and to infinity for p < 1/R. 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 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. Nonetheless, our proof is probabilistic and relies on a martingale hidden in the Faber-Schauder expansion of x. We show that the condition of our main result is satisfied for the typical sample paths of fractional Brownian motion with drift, and we provide almost sure convergence rates for the corresponding Gladyshev estimates. We also discuss the connections between the roughness exponent and the related concepts of Besov regularity and weighted quadratic variation. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators. Finally, we extend our results to the case in which the pth variation of x is defined over a sequence of unequally spaced partitions.