英文标题：
《The sum of log-normal variates in geometric Brownian motion》
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作者：
Ole Peters and Alexander Adamou
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最新提交年份：
2018
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英文摘要：
Geometric Brownian motion (GBM) is a key model for representing self-reproducing entities. Self-reproduction may be considered the definition of life [5], and the dynamics it induces are of interest to those concerned with living systems from biology to economics. Trajectories of GBM are distributed according to the well-known log-normal density, broadening with time. However, in many applications, what\'s of interest is not a single trajectory but the sum, or average, of several trajectories. The distribution of these objects is more complicated. Here we show two different ways of finding their typical trajectories. We make use of an intriguing connection to spin glasses: the expected free energy of the random energy model is an average of log-normal variates. We make the mapping to GBM explicit and find that the free energy result gives qualitatively correct behavior for GBM trajectories. We then also compute the typical sum of lognormal variates using Ito calculus. This alternative route is in close quantitative agreement with numerical work.
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中文摘要：
几何布朗运动（GBM）是表示自复制实体的关键模型。自我繁殖可能被认为是生命的定义，它所引发的动态对从生物学到经济学等生命系统的研究者都很感兴趣。GBM的轨迹按照众所周知的对数正态密度分布，随时间展宽。然而，在许多应用中，感兴趣的不是单个轨迹，而是多个轨迹的总和或平均值。这些物体的分布更加复杂。这里我们展示了两种不同的方法来找到它们的典型轨迹。我们利用了自旋玻璃的一个有趣的联系：随机能量模型的预期自由能是对数正态变量的平均值。我们明确地映射到GBM，并发现自由能结果给出了GBM轨迹的定性正确行为。然后，我们还使用伊藤演算计算对数正态变量的典型和。这条备选路线与数值计算结果在数量上非常一致。
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分类信息：

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：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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