New Problems in Stochastic Control Motivated by Mathematical Finance

粗略地说，博弈论的目的是确定双方在竞争中各自或共同所能做的最好的事情，在这种竞争中，每一方都试图实现一个目标，这个目标可能至少部分地与另一方的目标相矛盾。涉及两个以上参与方的游戏（竞赛）更为复杂，但对大规模游戏的分析提高了我们对金融、经济和工程等复杂系统的理解，否则这些系统是很难分析的。另一方面，提高对金融模型不确定性的理解有助于更好地管理风险。该研究项目探索这些领域的数学问题，旨在开发新的数学工具，灵感来自数学金融的应用。研究生和博士后研究人员直接参与这项工作。近年来，在金融和经济学的启发下，随机控制有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运输理论（即鞅最优运输）中的一些新问题。超级套期保值问题导致了几何动态规划原理的产生，而对大量参与者对整个系统影响很小的博弈的纳什均衡的分析导致了平均场博弈理论的产生。本研究项目旨在通过提供一些新的数学工具和研究一些由应用驱动的新问题来促进这些发展。该项目旨在通过模型不确定性、几何动态规划原理、随机控制的随机化方法、平均场型控制问题和平均场博弈进一步加深对金融数学的理解。

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.