New Problems in Stochastic Control Motivated by Mathematical Finance

点击翻译按钮获取中文摘要

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.