Topics in Mathematical Finance and Stochastic Control

数学金融与随机控制专题

• 批准号: 0604643
• 负责人: Mihai Sirbu
• 金额: $ 11.29万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-15 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604643&HistoricalAwards=false
• 关键词: Topics; Mathematical; Finance; Stochastic; Control

In this project, the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets and the sensitivity analysis of utility-based prices for the case of utility functions defined on the whole real line is considered. The research focuses on necessary and sufficient conditions under which the mathematical statements hold. In particular, we are looking for the largest class of contingent claims that can be analyzed in this framework. The risk-tolerance wealth process turns out to be the appropriate numeraire for pricing. In addition, a class of multi-dimensional stochastic games is studied.In complete markets, described by models like the Black and Scholes model, prices are uniquely defined by "no-arbitrage" considerations. In more realistic models, which are incomplete, prices of contingent claims can only be defined taking into account the preferences and wealth of specific investors. The dependence of prices on the number of contingent claims for investors who can borrow any amount of cash from the money market (infinite credit line) is studied. The pricing techniques can be applied to a large number of incomplete models, including options on non-traded assets and energy derivatives.

本文研究了不完全市场中最优投资问题中价值函数的二次可微性，以及效用函数定义在整条真实的直线上时基于效用的价格的灵敏度分析。 研究的重点是数学命题成立的充分必要条件。特别是，我们正在寻找可以在这个框架中分析的最大类别的或有债权。 风险容忍财富过程被证明是定价的合适的计算单位。此外，我们还研究了一类多维随机博弈，在完全市场中，如Black和Scholes模型，价格是由无套利因素唯一定义的。在更现实的不完整模型中，或有债权的价格只能在考虑特定投资者的偏好和财富的情况下确定。 研究了在无限信用额度条件下，投资者可以从货币市场上借入任意数量的现金时，价格对或有债权数量的依赖关系。 定价技术可以应用于大量的不完整模型，包括非交易资产和能源衍生品的期权。