Abstract

We present a modification of the Ramsey model that describes the consumer behavior of the households. We assume that the salary of the households is a stochastic process, defined by the stochastic differential equation (SDE). The impact of the large amount of the households can be modelled by a mean field term. This leads to a Kolmogorov–Fokker–Planck equation, evolving forward in time that describes the evolution of the probability density function of the households. Considering a Hamilton–Jacobi–Bellman equation, evolving backwards in time that describes the optimal strategy of the households behavior, we obtain a Mean Field Game problem. We present a self-similar solution of the Hamilton–Jacobi–Bellman equation and introduce the numerical solution of the Kolmogorov–Fokker–Planck equation.

中文翻译：

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基于平均场博弈方法的家庭行为建模

摘要

我们提出了描述家庭消费者行为的拉姆齐模型的修改。我们假设家庭的工资是一个随机过程，由随机微分方程（SDE）定义。大量家庭的影响可以用平均场项建模。这导致 Kolmogorov-Fokker-Planck 方程随着时间的推移而向前发展，它描述了家庭概率密度函数的演变。考虑到描述家庭行为最优策略的 Hamilton-Jacobi-Bellman 方程，在时间上向后演化，我们得到了一个平均场博弈问题。我们提出了 Hamilton-Jacobi-Bellman 方程的自相似解，并介绍了 Kolmogorov-Fokker-Planck 方程的数值解。