随机微分博弈模型中的库存管理问题：马尔可夫链近似和最优策略
Inventory Management Problems in Stochastic Differential Game Models: Markov Chain Approximation and Optimal Policies
摘要: 本文研究了在随机参考价格影响下，两个垄断厂商竞争下的生产和定价策略。设定的库存管理系统包括随机参考价格和随机需求。在随机微分博弈模型的框架下研究了库存管理问题，我们给出该背景下支付函数的定义。为了得到最优生产和定价，我们采用动态规划原理的方法，博弈的上下值满足一个耦合的非线性积分微分Hamilton-Jacobi-Isaacs (HJI)方程组。本文还证明了该对策问题鞍点的存在性，由于很难得到封闭形式的解，我们采用马尔可夫链近似来近似值函数和最优控制，并给出了收敛性分析。最后，我们进行了数值实验，并且根据实验结果，提出了相应的管理建议。
Abstract: This paper investigates the production and pricing strategies employed by two monopolies in a competitive environment, considering the influence of random reference prices. The inventory management system incorporates both random reference prices and random demand. The problem of inventory management is examined within the framework of a stochastic differential game model, with the payment function defined accordingly. To obtain optimal production and pricing decisions, dynamic programming principles are utilized, and coupled nonlinear integral differential Hamilton-Jacobi-Isaacs (HJI) equations that govern the upper and lower values of the game are established. Furthermore, the existence of saddle points for this game problem is proven. Due to challenges in obtaining closed-form solutions, Markov chain approximation is employed to approximate functions and optimal controls while providing convergence analysis. Finally, numerical experiments are conducted to validate our findings, leading to corresponding managerial recommendations.
文章引用：欧君恒, 卢相刚. 随机微分博弈模型中的库存管理问题：马尔可夫链近似和最优策略[J]. 应用数学进展, 2024, 13(4): 1827-1841. https://doi.org/10.12677/aam.2024.134172

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