摘要翻译：
本文研究了具有随机波动率的指数Ornstein-Uhlenbeck模型中金融收益概率分布的解析刻画问题。在该模型中，价格由一个几何布朗运动驱动，其扩散系数由一个均值回复过程控制的隐变量Y的指数函数表示。导出了两种极限情况下概率分布及其特征函数的闭式表达式。在第一种方法中，Y的波动大于波动率正常水平，而第二种方法对应于Y方差的小平稳值的假设。通过大量的蒙特卡罗模拟，对理论结果进行了数值检验。通过仔细分析欧拉-丸山格式数值实现中涉及的参数，检验了分析预测的有效性，并在金融指数数据集上进行了检验。特别地，我们讨论了德国DAX30和道琼斯欧洲斯托克50的结果，发现经验数据和理论描述之间有很好的一致性。
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英文标题：
《Probability distribution of returns in the exponential
  Ornstein-Uhlenbeck model》
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作者：
Giacomo Bormetti, Valentina Cazzola, Guido Montagna and Oreste
  Nicrosini
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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：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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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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一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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英文摘要：
  We analyze the problem of the analytical characterization of the probability distribution of financial returns in the exponential Ornstein-Uhlenbeck model with stochastic volatility. In this model the prices are driven by a Geometric Brownian motion, whose diffusion coefficient is expressed through an exponential function of an hidden variable Y governed by a mean-reverting process. We derive closed-form expressions for the probability distribution and its characteristic function in two limit cases. In the first one the fluctuations of Y are larger than the volatility normal level, while the second one corresponds to the assumption of a small stationary value for the variance of Y. Theoretical results are tested numerically by intensive use of Monte Carlo simulations. The effectiveness of the analytical predictions is checked via a careful analysis of the parameters involved in the numerical implementation of the Euler-Maruyama scheme and is tested on a data set of financial indexes. In particular, we discuss results for the German DAX30 and Dow Jones Euro Stoxx 50, finding a good agreement between the empirical data and the theoretical description.
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PDF链接：
https://arxiv.org/pdf/0805.0540