Measuring the roughness of random paths by increment ratios

TL;DR

This paper introduces an IR-based statistic for measuring the roughness of random paths, demonstrating its robustness and applicability to various stochastic processes, including diffusion, Gaussian, and Lévy processes.

Contribution

It defines the IR-roughness statistic, analyzes its properties, and explicitly computes it for specific classes of stochastic processes, extending roughness measurement methods.

Findings

IR-roughness coincides with Brownian roughness for diffusions

Explicit IR-roughness formulas for Gaussian processes

Estimation of stable process parameter using IR-roughness

Abstract

A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (called the IR-roughness below) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. Firstly, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Secondly, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Thirdly, the IR-roughness of a L\'evy process with $\alpha-$stable tangent process is established and can be used to estimate the fractional parameter $\alpha \in (0,2)$ following a central limit theorem.