Measuring the roughness of random paths by increment ratios

Abstract

A statistic based on increment ratios (IR’s) and related to zero crossings of an increment sequence is defined and studied for the purposes of 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 (which we shall call the IR-roughness) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. First, 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. Second, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Third, the IR-roughness of a Lévy process with an $α$-stable tangent process is established and can be used to estimate the fractional parameter $α ∈ (0, 2)$ following a central limit theorem.