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

随机博弈主题、模型不确定性控制问题以及金融应用

• 批准号: 1517664
• 负责人: Mihai Sirbu
• 金额: $ 28.92万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1517664&HistoricalAwards=false
• 关键词: Topics; stochastic; games; control; problems

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.

Sirbudms-1517664许多随机优化问题涉及多个玩家，通常具有竞争利益。 例如，一个球员的奖励可以是另一个玩家的成本，被称为两人零和游戏。 在正式层面上，可以类似地将所谓的“强大优化问题”模型，在最坏情况下，主动/智能玩家优化了奖励（由被动/无趣的玩家选择，被视为自然）。 该项目的重点是这种零和强大的优化问题，在这种问题上，没有预期的是具有纯粹策略的均衡（这意味着玩家不引入额外的随机性）。 研究了具有混合策略的零和游戏值的建模和价值。 预计，尽管具有类似的分析表示，但真正的零和游戏和强大的优化问题在存在混合策略的情况下的行为有所不同。 这些数学模型作为对不确定性时必须做出决定的问题的描述。他们的解决方案揭示了如何在替代方案中进行选择。 应用在工程以及金融和经济学领域发生。 另一个方向涉及金融经济学的问题：具有高水位标志性能费用的最佳投资策略。 学生被包括在项目的工作中。 该项目着重于没有ISAAC条件的游戏的建模和分析。 在具有两名活跃玩家的真正零和游戏中，混合策略的建模是不平凡的。 一种方法是允许在离散的时间网格上更改操作（包括混合），然后尝试找到游戏值。 看来，两个玩家都限制或不限制网格的情况会导致不同的结果。 对于强大的优化问题（在无趣的玩家选择开环控件的情况下），允许唯一的智能玩家随机化可能会导致更高的价值函数。 使用Perron方法的概率修改对动态编程分析特别注意。 该项目的第二个主题研究了一般的二维反映的最佳投资扩散模型，并具有性能费用。 最佳控制的反馈表示起着重要作用。 学生被包括在项目的工作中。