发表机构
Ceremade, CNRS-UMR 7534, Université Paris-Dauphine PSL; SPE CNRS-UMR 6134, Université de Corse（巴黎第九大学; 科西嘉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文对 Log S-fBM 随机波动率模型进行统计分析，推导标度性质、偏差不等式及假设检验，以区分粗糙与多重分形动态，并验证标度不变性。
AI 中文摘要
由 Wu 等人提出的 Log S-fBM 模型是一种随机波动率模型，其对数波动率是一个平稳分数布朗运动（S-fBM）：一个具有幂律衰减自协方差的平稳高斯过程，该自协方差由 Hurst 指数 $H$ 驱动，方差由间歇性系数缩放。一个关键性质是，它调和了粗糙波动率（其中 $H$ 通常接近 $0.1$，参见 Gatheral 等人）与多重分形波动率（其中 $H$ 接近 $0$，如 Bacry、Muzy 等人所述）：当 $H\ o0$ 时，模型的波动率测度收敛于一个多重分形随机测度。Wu 等人的数值研究显示，金融资产中的间歇性约为 $0.02$，这促使采用小间歇性近似对数波动率矩，以便通过广义矩方法（GMM）进行校准。在本工作中，我们对 Log S-fBM 模型进行了统计分析。我们推导了 S-fBM 过程和 Log S-fBM 积分波动率测度的标度性质，给出了对 $H$ 和间歇性敏感的尾分布的偏差不等式，并开发了一个针对零假设 Hurst 指数的假设检验，即粗糙动态与多重分形动态的判别。最后，我们通过显式的小间歇性公式重新审视了对数波动率增量过程的标度不变性，在两种机制中再现了类似的性质。
英文摘要
The Log S-fBM model, introduced by Wu et al., is a stochastic volatility model whose log volatility is a stationary fractional Brownian motion (S-fBM): a stationary Gaussian process with power-decaying autocovariance driven by the Hurst exponent $H$, and variance scaled by an intermittency coefficient. A key property is that it reconciles rough volatility, where $H$ is typically near $0.1$ (see Gatheral et al.), with multifractal volatility, where $H$ is close to $0$ as in Bacry, Muzy et al.: the model's volatility measure converges to a multifractal random measure as $H\to0$. Numerical findings in Wu et al. show intermittency of order $0.02$ across financial assets, motivating a small intermittency approximation of log volatility moments for calibration via the general method of moments (GMM). In this work, we conduct a statistical analysis of the Log S-fBM model. We derive scaling properties of the S-fBM process and the Log S-fBM integrated volatility measure, present deviation inequalities with tail distributions sensitive to $H$ and intermittency, and develop a hypothesis test for the null Hurst exponent, i.e.\ rough versus multifractal dynamics. Finally, we revisit scale invariance of the log volatility increment process via explicit small-intermittency formulas, reproducing analogous properties in both regimes.
