Topics in Financial Mathematics and Stochastic Control

金融数学与随机控制专题

• 批准号: 0908441
• 负责人: Mihai Sirbu
• 金额: $ 21.52万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0908441&HistoricalAwards=false
• 关键词: Topics; Financial; Mathematics; Stochastic; Control

The project funded by this award consists of a collection of topics in Stochastic Control and Financial Mathematics. The first topic is an optimal investment problem where the investor is paying a proportional share of the profit to the fund manager. The problem involves both modeling and the analysis of the resulting non-standard stochastic control problem, in order to fully characterize the optimal policy. In addition, an asymptotic analysis for small proportional fees will be performed. The second topic represents a characterization of semimartingale models of financial markets where Mutual Fund Theorems hold true in the context of expected utility from consumption. This will provide a tool for the study of mutual fund theorems for markets in equilibrium. The last topic is based on a new definition of admissible strategies for the optimal investment problem for utilities defined on the whole real line and the duality results that derive from it.Incomplete markets are financial models where contingent claims cannot be replicated by trading, so they are not redundant. While, in practice, most models are incomplete, the mathematical analysis of optimal investment and pricing in these markets is usually very difficult. The present project contributes to both modeling of incompleteness arising from different market frictions, as well as to the mathematical analysis of the new stochastic control problems resulting from such models. The second topic of the project is expected to provide a better understanding of incomplete markets in a very general mathematical framework.

由该奖项资助的项目包括一系列随机控制和金融数学的主题。 第一个主题是一个最佳投资问题，投资者将利润的比例份额向基金经理支付。 该问题既涉及建模和对所得非标准随机控制问题的分析，以便充分表征最佳策略。 另外，将对小比例费用进行渐近分析。 第二个主题代表了金融市场的半明星模型的特征，在该模型中，共同基金定理在消费的预期效用的背景下成立。 这将为研究平衡市场的共同基金定理提供工具。最后一个主题是基于对整个真实行定义的实用程序的最佳投资问题的新定义和从中定义的二元性结果的最佳投资问题的定义。完整的市场是财务模型，在这种模型中，不可用交易来复制偶然的主张，因此它们不是冗余的。在实践中，大多数模型都是不完整的，但这些市场中最佳投资和定价的数学分析通常非常困难。本项目既有助于不同的市场摩擦引起的不完整性建模，又是对这些模型引起的新随机控制问题的数学分析。预计该项目的第二个主题将在非常一般的数学框架中更好地了解不完整的市场。