New Problems in Stochastic Control Motivated by Mathematical Finance

数学金融引发的随机控制新问题

• 批准号: 1613170
• 负责人: Erhan Bayraktar
• 金额: $ 33.92万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613170&HistoricalAwards=false
• 关键词: New; Problems; Stochastic; Control; Motivated

Roughly speaking, game theory aims to determine the best that two parties can do, separately or together, in contests in which each attempts to achieve an objective that may be at least partially contradictory to that of the other. Games (contests) involving more than two parties are more complicated, but the analysis of large-population games improves our understanding of complex systems in finance, economics, and engineering that are otherwise difficult to analyze. On the other hand, improved understanding of model uncertainty in finance leads to a better management of risk. This research project explores mathematical questions in these areas and aims to develop new mathematical tools, inspired by applications in mathematical finance. Graduate students and post-doctoral researchers are directly involved in the work. There have been some exciting developments in stochastic control inspired by finance and economics in recent years: Financial modeling with model uncertainty led to some new problems in optimal transport theory (namely the martingale optimal transport). The super-hedging problems led to the geometric dynamic programming principle, and the analysis of Nash equilibria of games with a large number of players each having a very little influence on the overall system led to the theory of mean field games. This research project aims to contribute to these developments by providing some new mathematical tools and studying some new questions motivated by applications. The project aims to further advance understanding of financial mathematics with model uncertainty, the geometric dynamic programming principle, randomization approaches to stochastic control, and mean field type control problems and mean field games.

粗略地说，博弈论的目的是确定两方在比赛中单独或共同能够做到的最好成绩，在比赛中，每一方都试图实现与另一方至少部分矛盾的目标。涉及两方以上的博弈（竞赛）更为复杂，但对大规模博弈的分析可以提高我们对金融、经济和工程中难以分析的复杂系统的理解。另一方面，加深对金融模型不确定性的理解可以更好地管理风险。该研究项目探索这些领域的数学问题，旨在受数学金融应用的启发，开发新的数学工具。 研究生和博士后研究人员直接参与这项工作。近年来，受金融和经济学的启发，随机控制取得了一些令人兴奋的发展：具有模型不确定性的金融建模导致了最优传输理论（即鞅最优传输）中的一些新问题。超对冲问题催生了几何动态规划原理，而对大量参与者博弈的纳什均衡分析，每个参与者对整个系统的影响很小，催生了平均场博弈理论。该研究项目旨在通过提供一些新的数学工具并研究一些由应用引发的新问题，为这些发展做出贡献。该项目旨在进一步加深对具有模型不确定性的金融数学、几何动态规划原理、随机控制的随机化方法以及平均场类型控制问题和平均场博弈的理解。