Stochastic Control under Model Uncertainty

模型不确定性下的随机控制

• 批准号: 1208985
• 负责人: Marcel Nutz
• 金额: $ 13.16万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1208985&HistoricalAwards=false
• 关键词: Stochastic; Control; under; Model; Uncertainty

The problem of model uncertainty has recently received widespread attention in applied mathematics, especially after being identified as a cause for the recent credit crisis. The investigator seeks to develop probabilistic tools for stochastic control problems under uncertainty, in particular in mathematical finance and related game theoretic problems. The focus is on situations of severe model uncertainty, where no reference probability measure can be constructed and, therefore, standard tools from stochastic analysis cannot be applied. The first part of this project studies the problem of superhedging for a European, possibly path-dependent option under model uncertainty as well as the fundamental duality between trading strategies and (possibly singular) risk-neutral measures. The second part is concerned with stochastic differential games of control and stopping and, in a special case, the pricing of American options under uncertainty. The third part investigates a nonlinear version of the Doob-Meyer decomposition, which is a step towards a theory of semimartingales and stochastic integration under uncertainty. This project seeks a better mathematical understanding of model uncertainty. Its results help to resolve important questions of pricing, risk measurement, and hedging in mathematical finance. Moreover, the project leads to methodological developments in the theory of stochastic analysis and control.

模型不确定性问题最近在应用数学领域受到广泛关注，特别是在被确定为最近的信贷危机的原因之后。 研究人员寻求开发用于不确定性下的随机控制问题的概率工具，特别是在数学金融和相关博弈论问题中。 重点是模型严重不确定的情况，无法构建参考概率度量，因此无法应用随机分析的标准工具。 该项目的第一部分研究模型不确定性下欧洲可能依赖路径的期权的超级对冲问题，以及交易策略和（可能是单一的）风险中性措施之间的基本二元性。 第二部分涉及控制和停止的随机微分博弈，以及在特殊情况下不确定性下美式期权的定价。 第三部分研究了 Doob-Meyer 分解的非线性版本，这是朝着不确定性下的半鞅和随机积分理论迈出的一步。 该项目寻求对模型不确定性的更好的数学理解。 其结果有助于解决数学金融中的定价、风险测量和对冲等重要问题。 此外，该项目还导致了随机分析和控制理论的方法论发展。