一类零程过程的平稳分布收敛性及其在含债务财富分布模型中的应用
Convergence of stationary distributions for a class of zero-range processes and its application to a wealth distribution model with debt
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中文总结 AI 辅助
本文在含债务的代理人模型中引入随机交易机制，将其建模为Z值零程过程，证明平稳财富分布在适当缩放下收敛于双侧Gamma或Gamma分布，为理解随机货币交换产生宏观财富分布提供了数学框架。
中文摘要 AI 辅助
理解宏观财富分布如何从微观交易规则中涌现是经济物理学中的一个核心问题。虽然许多传统模型将代理人的财富限制为非负值，但债务是现实经济系统的基本特征。在本文中，我们将随机交易机制引入到具有中央银行中介的集体债务限额的基于代理人的模型中，如文献[14]所研究的那样。在我们的模型中，每个代理人根据其当前资产或债务水平，概率性地决定是否转移一枚硬币。从概率论的角度来看，所得系统可以表述为一个零程过程，其位点占用数为$\mathbb{Z}$值，其中负值代表债务。我们证明，在适当的缩放下，平稳状态下的财富分布收敛于双侧Gamma分布或Gamma分布，具体取决于控制硬币转移的函数的渐近行为。我们的结果严格识别了这些先前未被考虑的极限分布，并为理解随机货币交换如何产生宏观财富分布提供了数学框架。
英文摘要
Understanding how macroscopic wealth distributions emerge from microscopic transaction rules among agents is a central problem in econophysics. While many traditional models restrict agents' wealth to non-negative values, debt is an essential feature of realistic economic systems. In this paper, we introduce a stochastic transaction mechanism into the agent-based model with a central-bank-mediated collective debt limit, as studied in [14]. In our model, each agent probabilistically determines whether to transfer a coin, depending on their current asset or debt level. From a probabilistic perspective, the resulting system can be formulated as a zero-range process with $\mathbb{Z}$-valued site occupation numbers, where negative values represent debt. We prove that, under an appropriate scaling, the wealth distribution in the stationary state converges to either a two-sided Gamma distribution or a Gamma distribution, depending on the asymptotic behavior of the function governing coin transfers. Our results rigorously identify these previously unconsidered limiting distributions and provide a mathematical framework for understanding how stochastic monetary exchanges give rise to macroscopic wealth distributions.
发表机构
- Keio University（庆应义塾大学）
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