Equilibrium with Randomized Strategies in Learning Theory and Mathematical Finance

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The objective of the research project is to contribute to the understanding of the interaction between agents that might have contradictory interests. The project addresses two topics where the optimal strategies of the agents are obtained via particular randomization. The first topic is in financial mathematics and concerns the asymmetry of information in financial markets. In this framework, the main question of interest is to understand how the agents interact and how the price processes evolve if some of the agents have superior or inferior information. The research project is expected to lead to novel tools of risk management in financial markets. The second topic is in learning theory and considers the interaction between an agent (the learner) who aims to predict the outcome of future events based on additional information on these events and an adversary who aims to make the task of the learner as difficult as possible. In this prediction with expert advice framework, the objective of the investigator is to derive optimal learning strategies for the learner. Graduate students are involved in the project.To be more specific, regarding the first topic, the investigator will establish the existence of equilibrium in financial markets with long-lived asymmetric information. Unlike the classical formulation of the problem via Hamilton-Jacobi-Bellman equations, the investigator will find an equilibrium by using tools from convex analysis and optimal transport. Then, the properties of the equilibrium strategies and pricing rules will be studied in various extensions of the problem such as stochastic liquidity and risk-averse agents with natural distributional assumptions. For the second topic, the interaction between the learner and the adversary will be stated as a zero-sum stochastic game. Then, the long-time behavior of these games will be studied using tools from partial differential equations, stochastic analysis, and mean-field theory. An important objective will be to find simple characterizations of asymptotic Nash equilibria and to assess the performances of classical learning algorithms.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.