Many questions in finance, economics, and engineering involve competition or interaction among a large number of agents. Large-population games have traditionally been difficult to analyze due to the large size of the system and the game nature of the interaction. The theory of mean field games provides a useful way to approximate these complex systems, and the theory helps to improve understanding of the effects of interactions and how populations react to different compensation schemes or public policies. On the other hand, the 2008 financial crisis revealed the fragility of financial models. Since then, the urge to develop financial theories that take into account model risk has grown tremendously. This research project explores mathematical questions in mean field game theory and robust finance.
The first part of the project analyzes a dynamic competition involving a large number of players, where the interaction among players is through the ranking of the completion time of their respective projects. The model applies to situations where many firms or individuals compete to be the first to achieve a goal. The objective is to understand the equilibrium and design a reward scheme that encourages early project completion given that the organizer has a limited budget, or to minimize the budget given a desired rate of completion. The second part of the project is concerned with the accuracy of the mean field game approximation, when in many applications, the typical size of competition is only modestly large. The objective is to study the fluctuation around the hydrodynamic limit of the mean field game approximation so as to improve accuracy. The third part of the project studies a mean field game of optimal stopping when the interaction among players is neither through the state process nor the cost structure, at least not in a direct way, but through the belief or information revealed from the action of stopping. The last part of the project considers pricing and hedging of contingent claims in a financial market under both transaction costs and model uncertainty, where model uncertainty is described by a collection of probability measures. In general the collection need not have a reference measure that dominates every other measure, and therefore, standard tools from functional analysis cannot be applied, and new techniques are called for.