Topics in stochastic games, control problems with model uncertainty and applications to finance

University Of Texas At Austin, Austin TX

Investigators

Abstract

Sirbu DMS-1517664 Many stochastic optimization problems involve more than one player, often having competing interests. For example, one player's reward can be the other player's cost, known as a two-person zero-sum game. At a formal level, one can similarly model so called robust optimization problems, where an active/intelligent player optimizes a reward under the worst case scenario (chosen by a passive/uninterested player, thought of as nature). The project focuses on such zero-sum games and robust optimization problems where the existence of equilibria with pure strategies (which means that players do not introduce additional randomness) is not expected. The modeling and existence of a value for zero-sum games with mixed strategies is studied. It is expected that, despite a similar analytic representation, genuine zero-sum games and robust optimization problems behave differently in the presence of mixed strategies. These mathematical models arise as descriptions of problems in which one must make decisions in the face of uncertainty; their solutions reveal how to choose among alternatives. Applications occur in areas of engineering as well as finance and economics. Another direction concerns a problem in Financial Economics: optimal investment strategies with high-watermark performance fees. Students are included in the work of the project. The project focuses on the modeling and analysis of games without Isaacs conditions. In a genuine zero-sum game with two active players, modeling of mixed strategies is non-trivial. One way is to allow for actions (including mixing) to be changed over discrete time grids and then attempt to find a value for the game. It appears that the cases when both players are restricted or not to the same time grid lead to different results. For a robust optimization problem (where the uninterested player chooses open-loop controls), allowing the only intelligent player to randomize may lead to a better value function. Special attention is given to the dynamic programming analysis using probabilistic modifications of Perron's method. A second topic of the project studies a general two-dimensional reflected diffusion model of optimal investment with performance fees. The feedback representation of the optimal control plays a prominent role. Students are included in the work of the project.