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

许多随机优化问题涉及不止一个参与者，通常有竞争利益。例如，一个玩家的奖励可能是另一个玩家的成本，这就是所谓的两人零和游戏。在正式层面上，我们可以类似地模拟所谓的稳健优化问题，即主动/智能玩家在最坏情况下优化奖励（由被动/不感兴趣的玩家选择，被认为是自然）。该项目专注于零和博弈和鲁棒优化问题，其中不存在纯策略均衡（这意味着玩家不引入额外的随机性）。研究了混合策略零和博弈的建模和存在性问题。我们期望，尽管有类似的分析表示，真正的零和博弈和鲁棒优化问题在混合策略的存在下表现不同。这些数学模型是对问题的描述，在这些问题中，人们必须面对不确定性做出决定；他们的解决方案揭示了如何在备选方案中进行选择。应用出现在工程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：具有高水位绩效费的最优投资策略。学生们被包括在这个项目的工作中。该项目侧重于对没有艾萨克条件的游戏进行建模和分析。在一个真正的零和游戏中，有两个积极的参与者，混合策略的建模是非常重要的。一种方法是允许行动（包括混合）在离散的时间网格中改变，然后尝试为游戏找到一个值。当两名玩家都被限制在相同的时间网格中时，结果似乎是不同的。对于一个稳健的优化问题（不感兴趣的玩家选择开环控制），允许唯一聪明的玩家随机选择可能会产生更好的价值函数。特别注意了利用Perron方法的概率修正进行动态规划分析。项目的第二个课题研究了考虑绩效费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。学生们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.