Problems in Mathematical Finance

数学金融问题

• 批准号: 1411809
• 负责人: Kasper Larsen
• 金额: $ 14.58万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411809&HistoricalAwards=false
• 关键词: Problems; Mathematical; Finance

LarsenDMS-1411809 In the first part of this project, the investigator and his colleagues seek to use methods from stochastic analysis to provide a comprehensive study of price formation in financial markets such as the stock and fixed-income markets. The specific goal is to understand how idiosyncratic risks affect equilibrium price formation. Currently, there are very few results available that can answer such questions and this part of the project seeks to provide tractable models that can be used to obtain approximate answers. In the second part, the investigator and his colleagues seek to develop optimization tools that can deal with a large class of stochastic control problems often encountered in mathematical finance. These problems exhibit unexpected discontinuities that prevent the standard mathematical tools from being applicable. From a rigorous mathematical perspective, the investigator first seeks to establish the existence of incomplete equilibria in continuous-time models governed by Brownian motions. Such models are well-known for being notoriously intractable. Therefore, the investigator seeks to provide tractable approximation tools that can be used as surrogates for the general models. Secondly, the investigator and his colleagues seek to provide a partial differential equation characterization of the problem of optimal investment with unspanned endowment. This control problem turns out to have a discontinuous value function (a facelift or boundary layer), which prevents the use of standard partial differential equation techniques. Finally, the investigator and his colleagues seek to develop tools that can detect up front such discontinuities for a general class of stochastic control problems.

拉森DMS-1411809 在这个项目的第一部分，研究者和他的同事试图使用随机分析的方法，提供一个全面的研究价格形成的金融市场，如股票和固定收益市场。 具体目标是了解特殊风险如何影响均衡价格形成。 目前，可以回答这些问题的结果很少，项目的这一部分旨在提供可用于获得近似答案的易处理模型。 在第二部分中，研究者和他的同事们寻求开发优化工具，可以处理数学金融中经常遇到的一大类随机控制问题。 这些问题表现出意想不到的不连续性，使标准的数学工具无法应用。 从严格的数学角度来看，研究者首先试图建立由布朗运动控制的连续时间模型中不完全平衡的存在性。 众所周知，这种模型非常难以处理。 因此，研究人员试图提供易于处理的近似工具，可以用作一般模型的替代品。 其次，研究者和他的同事们试图提供一个偏微分方程刻画的问题的最优投资与非跨越禀赋。 这个控制问题原来有一个不连续的值函数（整容或边界层），这阻止了使用标准的偏微分方程技术。 最后，研究人员和他的同事们寻求开发工具，可以检测到前面这样的不连续性的一般类的随机控制问题。