Topics in Stochastic Control and Games Motivated by Finance

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Many real-life situations (investing in financial markets, modeling traffic, etc.) involve decision making under uncertainty. Occasionally, the decision maker can even face an opposing player, informed or not, leading to a so-called zero-sum game. Modeling and analysis of games is very difficult, in large part because of the potential strategic behavior of the two players. This project studies the theory behind a collection of topics in stochastic control (one decision maker) or games (two opposing players). Some of the models are directly motivated by finance, but the general theory could be applied to other situations. The first topic aims to provide a better explanation of how an investor should adjust their investing strategies (proportion in each asset) in the presence of small transaction costs, compared to not facing such costs. The second topic is a general treatment of stochastic games, with the aim to explain the difference between having an intelligent opponent (another person, acting strategically) and an uninformed one, like nature choosing a worst-case scenario for the decision maker. The third topic is of more technical nature, advancing the mathematical theory behind a peculiar kind of fees that investor can face, namely performance fees.The present project aims to study three topics in stochastic control and games. The first is a study of multi-dimensional singular control problems, with the main application to approximation of optimal investment strategies with small proportional transaction costs. The goal is to deeper understand the structure of feedback optimal strategies as solutions to reflected stochastic differential equations. The second topic will study the difference in both modeling and analysis between genuine zero-sum games and stochastic control under model uncertainty. While, analytically, these two situations appear to be described by the same Isaacs equation, if the Isaacs condition does not hold, allowing the player to use randomized strategies may lead to different models (and values). Finally, the PI will study the duality theory for investing with performance fees. Preliminary investigation shows that the dual problem is a reflected two-dimensional diffusion.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.