摘要翻译：
我们研究了美国股票市场1137只股票在2001-2002年间的波动时间序列，并分析了它们的回报间隔$\tau$,即波动超过给定阈值$q$之间的时间间隔。我们研究了$\tau$,$P_q(\tau)$的概率密度函数，假设一个伸展指数函数$P_q(\tau)\sim e^{-\tau^\gamma}$。我们发现指数$\γ$依赖于波动率在$q=1$～6个标准差范围内的阈值。这一发现支持了返回间隔分布的多尺度性质。为了更好地理解多尺度的起源，我们研究了$\γ$如何依赖于四个基本因素，资本化，风险，交易数量和回报。我们发现$\γ$依赖于资本化、风险和收益，但几乎不依赖于交易数量。这表明$\γ$与投资组合选择有关，而与市场活动无关。为了进一步刻画个股的多尺度性，我们对$\tau$,$\mu_m\equiv<(\tau/<\tau>)^m>^{1/m}$在$10<<\tau>\le100美元范围内的矩进行了幂律拟合，$\mu_m\sim<\tau>^\delta$。指数$\delta也与资本化、风险和收益有关，但与交易次数无关，其趋势与$\gamma$相反。此外，我们还证明了$\delta$与$\gamma$近似成线性关系地递减。收益区间显示了波动的时间结构，我们的发现表明它们的多尺度特征可能有助于投资组合优化。
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英文标题：
《Multifactor Analysis of Multiscaling in Volatility Return Intervals》
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作者：
Fengzhong Wang, Kazuko Yamasaki, Shlomo Havlin and H. Eugene Stanley
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最新提交年份：
2008
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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一级分类：Physics        物理学
二级分类：Physics and Society        物理学与社会
分类描述：Structure, dynamics and collective behavior of societies and groups (human or otherwise). Quantitative analysis of social networks and other complex networks. Physics and engineering of infrastructure and systems of broad societal impact (e.g., energy grids, transportation networks).
社会和团体（人类或其他）的结构、动态和集体行为。社会网络和其他复杂网络的定量分析。具有广泛社会影响的基础设施和系统（如能源网、运输网络）的物理和工程。
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英文摘要：
  We study the volatility time series of 1137 most traded stocks in the US stock markets for the two-year period 2001-02 and analyze their return intervals $\tau$, which are time intervals between volatilities above a given threshold $q$. We explore the probability density function of $\tau$, $P_q(\tau)$, assuming a stretched exponential function, $P_q(\tau) \sim e^{-\tau^\gamma}$. We find that the exponent $\gamma$ depends on the threshold in the range between $q=1$ and 6 standard deviations of the volatility. This finding supports the multiscaling nature of the return interval distribution. To better understand the multiscaling origin, we study how $\gamma$ depends on four essential factors, capitalization, risk, number of trades and return. We show that $\gamma$ depends on the capitalization, risk and return but almost does not depend on the number of trades. This suggests that $\gamma$ relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of $\tau$, $\mu_m \equiv <(\tau/<\tau>)^m>^{1/m}$, in the range of $10 < <\tau> \le 100$ by a power-law, $\mu_m \sim <\tau>^\delta$. The exponent $\delta$ is found also to depend on the capitalization, risk and return but not on the number of trades, and its tendency is opposite to that of $\gamma$. Moreover, we show that $\delta$ decreases with $\gamma$ approximately by a linear relation. The return intervals demonstrate the temporal structure of volatilities and our findings suggest that their multiscaling features may be helpful for portfolio optimization.
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PDF链接：
https://arxiv.org/pdf/0808.3200