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

SirbuDMS-1517664许多随机优化问题涉及多个参与者，往往具有相互竞争的利益。例如，一个玩家的奖励可以是另一个玩家的成本，这就是所谓的两人零和博弈。在形式层面上，人们可以类似地建模所谓的稳健优化问题，其中主动/智能玩家在最坏的情况下优化奖励(由被动/不感兴趣的玩家选择，被认为是自然的)。该项目专注于这样的零和博弈和稳健优化问题，其中不期望存在纯策略的均衡(这意味着参与者不会引入额外的随机性)。研究了具有混合策略的零和对策的模型及其解的存在性。尽管有类似的解析表示，但预计真正的零和博弈和稳健优化问题在存在混合策略时的行为不同。这些数学模型是对一个人在面对不确定性时必须做出决定的问题的描述；它们的解决方案揭示了如何在备选方案中进行选择。应用领域包括工程领域以及金融和经济领域。另一个方向涉及金融经济学中的一个问题：高绩效费用的最优投资策略。学生被包括在这个项目的工作中。该项目专注于对没有艾萨克斯条件的游戏进行建模和分析。在一个有两个活跃参与者的真正零和博弈中，混合策略的建模并不是微不足道的。一种方法是允许在离散的时间网格上改变动作(包括混合)，然后尝试找到游戏的值。当两个玩家被限制或不被限制在同一时间网格时，似乎会导致不同的结果。对于稳健优化问题(不感兴趣的玩家选择开环控制)，允许唯一的智能玩家随机化可能会产生更好的值函数。特别注意使用Perron方法的概率修正的动态规划分析。该项目的第二个主题研究了具有绩效费用的最优投资的一般二维反射扩散模型。最优控制的反馈表示起着突出的作用。学生被包括在这个项目的工作中。

SirbuDMS-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.