用增量比测量随机路径的粗糙度
Measuring the roughness of random paths by increment ratios
摘要：
定义并研究了一种基于增量比（IR）的与增量序列过零点有关的统计量，用于测量随机路径的粗糙度。该统计量的主要优点是对平滑加性和乘性趋势的鲁棒性以及对无穷方差过程的适用性。IR统计极限（以下称为IR粗糙度）的存在与切过程的存在密切相关。三个特殊的情况下，IR粗糙度存在，并明确计算被认为是。首先，对于具有光滑扩散系数和漂移系数的扩散过程，IR-粗糙度与布朗运动的IR-粗糙度一致，并得到其收敛速度。其次，在一般假设下，详细研究了粗糙高斯过程的情况，这些假设不需要平稳性条件。第三，建立了具有$α-$稳定切过程的Lévy过程的IR-粗糙度，并根据中心极限定理，利用IR-粗糙度估计分数阶参数$α\in (0,2)$.
英文摘要：
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 Lévy process with $α-$stable tangent process is established and can be used to estimate the fractional parameter $α\in (0,2)$ following a central limit theorem.
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