In this paper, we consider the wealth distribution obtained by trading (buying–selling) stocks whose prices follow the geometric Brownian motion (GBM), when both number of the ticker symbol of the stock and maximum number of the traded stock are limited to unity. The binary exchange of the cash and stock between two agents is expressed with the Boltzmann-type kinetic equation. The distribution function of the number of the agents with the specific number of the stock or specific amount of the cash can be demonstrated, theoretically, when the price of the stock is constant. The distribution function of the number of the agents with the specific amount of the total asset can be approximated by Γ-distribution, when the price of the stock follows the GBM. Finally, the rule in the binary-exchange-game approximates the distribution function of the number of the agents with the specific amount of the total asset to the Feller–Pareto-like distribution at the high wealth tail.

中文翻译：

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用几何布朗运动交易股票的财富分配动力学模型

在本文中，我们考虑通过交易（买入-卖出）价格遵循几何布朗运动（GBM）的股票获得的财富分配，当股票的股票代码数量和交易股票的最大数量都限制为 1 时. 两个代理之间现金和股票的二元交换用玻尔兹曼型动力学方程表示。在股票价格不变的情况下，理论上可以证明具有特定数量的股票或特定数量的现金的代理人数量的分布函数。具有特定总资产数量的代理人数量的分布函数可以近似为Γ-分布，当股票价格遵循 GBM 时。最后，二元交换博弈中的规则将具有特定总资产数量的代理人数量的分布函数近似为高财富尾处的 Feller-Pareto 类分布。