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.

该奖项资助的项目包括随机控制和金融数学的主题集合。 第一个主题是一个最优投资问题，投资者支付一定比例的利润给基金经理。 该问题涉及建模和分析所产生的非标准随机控制问题，以充分表征最优策略。 此外，还将对小比例费用进行渐进分析。 第二个主题是金融市场的半鞅模型的特征，共同基金定理在消费的预期效用的背景下成立。 这将为研究均衡市场的共同基金定理提供一个工具。最后一个主题是基于对定义在整条真实的线上的公用事业的最优投资问题的容许策略的一个新的定义和由此得到的对偶结果。不完全市场是金融模型，其中或有债权不能通过交易复制，因此它们不是多余的。然而，在实践中，大多数模型是不完整的，在这些市场中的最优投资和定价的数学分析通常是非常困难的。本项目有助于两个模型的不完全性所产生的不同的市场摩擦，以及数学分析的新的随机控制问题所产生的这些模型。该项目的第二个主题预计将在一个非常一般的数学框架内提供对不完全市场的更好理解。