| 虞嘉禾,汤善健.噪声可退化且依赖于状态和分布的平均场博弈[J].数学年刊A辑,2020,41(3):233~262 |
| 噪声可退化且依赖于状态和分布的平均场博弈 |
| Mean-Field Game with Degenerate State- and Distribution-Dependent Noises |
| Received:January 07, 2020 Revised:May 23, 2020 |
| DOI：10.16205/j.cnki.cama.2020.0017 |
| 中文关键词: 平均场博弈, McKean-Vlasov型正倒向随机微分方程, 混沌传播, 随机最大值原理 |
| 英文关键词:Mean-Field games, McKean-Vlasov forward-backward stochastic differential equations,Propagation of chaos, Stochastic maximum principle |
| 基金项目:国家自然科学基金 (No.,11631004)和国家重点研发计划 (No.,2018YFA0703900) |
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| 中文摘要: |
| 文章考虑状态方程关于状态和控制仿射,效用关于状态和控制凸的平均场博弈, 允许状态方程的扩散项可退化且依赖状态和分布.由于允许漂移项和扩散项关于分布可以线性增长, 因此可以包含线性二次平均场博弈,且允许状态的期望以线性形式出现在状态方程中. 作者证明了对应的McKean-Vlasov型正倒向微分方程解的存在性,并获得了对应的解耦函数的正则性.最后作者证明了用平均场博弈的解和解耦函数可以以N^{-{1over d+4}}的速度逼近多人博弈的Nash均衡. |
| 英文摘要: |
| The mean-field game is studied for state-affine systems with degenerate state- and distribution-dependent noises. The mean field terms of both drift and diffusion coefficients are allowed to grow in the distribution in a linear way, and therefore the linear-quadratic case (where the expected state appears in a linear way in the system dynamics) is included. The authors prove existence of the solution to the associated forward-backward stochastic differential equations of a McKean-Vlasov type and regularity of the decoupled function. Finally, they prove that solutions of the mean-field game together with the decoupled function approximate the Nash equilibrium of the N-players' game with an order up to N^{-{1over d+4}}. |
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