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摘要翻译：
自Bachelier于1900年首次提出金融指数的演化模型以来，将其作为一个随机过程进行建模一直是一个有待全面、令人满意的解决方案的问题。本文指出，从历史序列中提取的回归概率密度函数随时间的缩放表明了一个成功的模型。由此产生的随机过程是一个异方差的非马尔可夫鞅，它可以用来模拟指数在自回归策略的基础上的演化。结果完全符合波动率聚类和回报分布的多尺度性质。将过程构造建立在标度基础上的思想及其构造本身受到统计力学的概率重整化群方法和强相关随机变量和的中心极限定理的最新表述的密切启发。
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英文标题：
《Role of scaling in the statistical modeling of finance》
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作者：
Attilio L. Stella and Fulvio Baldovin
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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：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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英文摘要：
Modeling the evolution of a financial index as a stochastic process is a problem awaiting a full, satisfactory solution since it was first formulated by Bachelier in 1900. Here it is shown that the scaling with time of the return probability density function sampled from the historical series suggests a successful model. The resulting stochastic process is a heteroskedastic, non-Markovian martingale, which can be used to simulate index evolution on the basis of an auto-regressive strategy. Results are fully consistent with volatility clustering and with the multi-scaling properties of the return distribution. The idea of basing the process construction on scaling, and the construction itself, are closely inspired by the probabilistic renormalization group approach of statistical mechanics and by a recent formulation of the central limit theorem for sums of strongly correlated random variables.
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PDF链接：
https://arxiv.org/pdf/0804.0331
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