Topics in Stochastic Control and Games Motivated by Finance

金融驱动的随机控制和博弈主题

• 批准号: 1908903
• 负责人: Mihai Sirbu
• 金额: $ 32.06万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1908903&HistoricalAwards=false
• 关键词: Topics; Stochastic; Control; Games; Motivated

Many real-life situations (investing in financial markets, modeling traffic, etc.) involve decision making under uncertainty. Occasionally, the decision maker can even face an opposing player, informed or not, leading to a so-called zero-sum game. Modeling and analysis of games is very difficult, in large part because of the potential strategic behavior of the two players. This project studies the theory behind a collection of topics in stochastic control (one decision maker) or games (two opposing players). Some of the models are directly motivated by finance, but the general theory could be applied to other situations. The first topic aims to provide a better explanation of how an investor should adjust their investing strategies (proportion in each asset) in the presence of small transaction costs, compared to not facing such costs. The second topic is a general treatment of stochastic games, with the aim to explain the difference between having an intelligent opponent (another person, acting strategically) and an uninformed one, like nature choosing a worst-case scenario for the decision maker. The third topic is of more technical nature, advancing the mathematical theory behind a peculiar kind of fees that investor can face, namely performance fees.The present project aims to study three topics in stochastic control and games. The first is a study of multi-dimensional singular control problems, with the main application to approximation of optimal investment strategies with small proportional transaction costs. The goal is to deeper understand the structure of feedback optimal strategies as solutions to reflected stochastic differential equations. The second topic will study the difference in both modeling and analysis between genuine zero-sum games and stochastic control under model uncertainty. While, analytically, these two situations appear to be described by the same Isaacs equation, if the Isaacs condition does not hold, allowing the player to use randomized strategies may lead to different models (and values). Finally, the PI will study the duality theory for investing with performance fees. Preliminary investigation shows that the dual problem is a reflected two-dimensional diffusion.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.

许多现实生活中的情况（在金融市场上进行投资，建模流量等）涉及在不确定性下做出决策。有时，决策者甚至可能会面对一个对立的球员，无论是否通知，都会导致所谓的零和游戏。对游戏的建模和分析非常困难，在很大程度上是由于两个玩家的潜在战略行为。该项目研究了随机控制（一个决策者）或游戏（两个对立的玩家）的主题集合背后的理论。其中一些模型是直接是由金融动机的，但是一般理论可以应用于其他情况。第一个主题旨在更好地解释投资者应如何在较小的交易成本的情况下调整其投资策略（每个资产的比例），而不是面临此类费用。第二个主题是对随机游戏的一般待遇，目的是解释拥有聪明的对手（另一个人，策略性地行事）和一个不知情的对手之间的区别，例如自然选择决策者的最糟糕场景。第三个主题是更具技术性的性质，推进了投资者可以面对的特殊费用背后的数学理论，即绩效费。目前的项目旨在研究随机控制和游戏中的三个主题。首先是对多维奇异控制问题的研究，主要应用于以较小的比例交易成本的最佳投资策略的近似。目的是更深入地了解反馈最佳策略的结构，作为反映随机微分方程的解决方案。第二个主题将研究真正的零和游戏的建模和分析的差异，以及模型不确定性下的随机控制。虽然在分析上，这两种情况似乎是由同一ISAACS方程描述的，如果ISAACS条件不存在，则允许玩家使用随机策略可能会导致不同的模型（和值）。最后，PI将研究二元性理论，用于绩效费用。初步调查表明，双重问题是一个反映的二维扩散。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准通过评估来支持的。