ABSTRACT This paper concerns the mathematical analysis of a mathematical model for price formation. We take a large number of rational buyers and vendors in the market who are trading the same good into consideration. Each buyer or vendor will choose his optimal strategy to buy or sell goods. Since markets seldom stabilize, our model mimics the real market behavior. We introduce three models. All of them are modifications of the original J.-M. Lasry and P. L. Lions evolution model. In the first modified model, a random term is added to mimic the randomness of trading in the real market. This reflects markets with low volatility, where it might be difficulty to buy or sell goods at specific price. In the second model, we use cumulative density function instead of density function. We give numerical simulations on these two models in order to have a general picture on the solution. In the third model, we add a term associated with the parameter R to destabilize the original Larsy–Lions model and study oscillations and wave solutions depending on different values of R. We also study existence and uniqueness of the solution. Moreover, Several plots are given to demonstrate these results corresponding to the theoretical prediction.

中文翻译：

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具有零诺依曼边界条件的价格形成模型

摘要 本文涉及价格形成数学模型的数学分析。我们考虑了市场上大量交易相同商品的理性买家和卖家。每个买家或卖家都会选择他的最佳策略来买卖商品。由于市场很少稳定，我们的模型模拟了真实的市场行为。我们介绍三个模型。所有这些都是对原始 J.-M 的修改。Lasry 和 PL Lions 进化模型。在第一个修改后的模型中，添加了一个随机项来模拟真实市场中交易的随机性。这反映了波动性较低的市场，可能难以以特定价格买卖商品。在第二个模型中，我们使用累积密度函数而不是密度函数。我们对这两个模型进行数值模拟，以便对解决方案有一个大致的了解。在第三个模型中，我们添加了一个与参数 R 相关的项来破坏原始 Larsy-Lions 模型的稳定性，并根据 R 的不同值研究振荡和波动解。我们还研究了解的存在性和唯一性。此外，给出了几个图来证明与理论预测相对应的这些结果。