In science and especially in economics, agent-based modeling has become a widely used modeling approach. These models are often formulated as a large system of difference equations. In this study, we discuss two aspects, numerical modeling and the probabilistic description for two agent-based computational economic market models: the Levy–Levy–Solomon model and the Franke–Westerhoff model. We derive time-continuous formulations of both models, and in particular, we discuss the impact of the time-scaling on the model behavior for the Levy–Levy–Solomon model. For the Franke–Westerhoff model, we proof that a constraint required in the original model is not necessary for stability of the time-continuous model. It is shown that a semi-implicit discretization of the time-continuous system preserves this unconditional stability. In addition, this semi-implicit discretization can be computed at cost comparable to the original model. Furthermore, we discuss possible probabilistic descriptions of time-continuous agent-based computational economic market models. Especially, we present the potential advantages of kinetic theory in order to derive mesoscopic descriptions of agent-based models. Exemplified, we show two probabilistic descriptions of the Levy–Levy–Solomon and Franke–Westerhoff model.

中文翻译：

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基于代理的计算经济市场模型的稳健数学公式和概率描述

在科学领域，尤其是在经济学领域，基于代理的建模已成为一种广泛使用的建模方法。这些模型通常被表述为一个大型差分方程系统。在这项研究中，我们讨论了两个方面，数值建模和两个基于代理的计算经济市场模型的概率描述：Levy-Levy-Solomon 模型和 Franke-Westerhoff 模型。我们推导出两个模型的时间连续公式，特别是，我们讨论了时间尺度对 Levy-Levy-Solomon 模型的模型行为的影响。对于 Franke-Westerhoff 模型，我们证明原始模型中所需的约束对于时间连续模型的稳定性不是必需的。结果表明，时间连续系统的半隐式离散化保持了这种无条件的稳定性。此外，这种半隐式离散化可以以与原始模型相当的成本计算。此外，我们讨论了基于时间连续代理的计算经济市场模型的可能概率描述。特别是，我们提出了动力学理论的潜在优势，以便推导出基于代理的模型的细观描述。例如，我们展示了 Levy-Levy-Solomon 和 Franke-Westerhoff 模型的两种概率描述。