| 郝涛.平均场型的随机微分对策和路径依赖的Bellman-Isaacs主方程[J].数学年刊A辑,2024,45(4):373~400 |
| 平均场型的随机微分对策和路径依赖的Bellman-Isaacs主方程 |
| Stochastic Differential Games of Mean-field Type and Path-dependent Bellman-Isaacs Master Equations |
| Received:March 16, 2024 Revised:November 19, 2024 |
| DOI：10.16205/j.cnki.cama.2024.0026 |
| 中文关键词: 随机微分对策 路径依赖的平均场随机微分方程 Bellman-Isaacs主方程 动态规划 粘性解 |
| 英文关键词:Stochastic differential game, Path-dependent mean-field SDE,
Bellman-Isaacs master equation, Dynamic programming principle,
Viscosity solution |
| 基金项目:国家自然科学基金(No.72171133)； 山东省自然科学基金(Nos.ZR2024MA039,ZR2022MA029) |
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| 中文摘要: |
| 作者研究由路径依赖平均场随机微分方程驱动的零和随机微分对策.动力系统和价值泛函的系数均依赖于解的路径以及路径的分布.采用典则场景下的非预期策略对反馈控制的弱解框架,因此值函数定义在平方可积的概率测度空间上.通过调整测试函数空间使其具备必要的局部紧性,给出路径依赖Bellman-Isaacs主方程粘性解的内生性定义.证明值函数的正则性和动态规划原理,从而得到相关的Bellman-Isaacs主方程解的一种概率解释. |
| 英文摘要: |
| The purpose of this paper is to analyse zero-sum stochastic differential games(SDGs for short), which is governed by path-dependent mean-field stochastic differentialequations (SDEs for short). All the coefficients of dynamics and cost functional depend onthe path of state and the distribution of path. Since the author uses a weak formulation withnonanticipative strategies vs feedback controls, value functions are defined on the space of allthe square integrable probability measures. By modifying the set of test functions to guarantee the necessary compactness, the author proposes an intrinsic notion of viscosity solutionsto path-dependent Bellman-Isaacs master equations. The regularity of value functions anddynamic programming principle are proved, and thereby the probabilistic interpretation forrelated Bellman-Isaacs master equations is obtained |
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