Stochastic Representation Problems for Optimal Control and Nonlinear Models for Illiquid Financial Markets

该项目专注于解决数学、金融和经济学中出现的随机优化问题的新方法。一个关键的主题是为非流动性金融市场开发和分析新的数学模型，这些模型解释了资产价格动态和交易策略之间的相关性，导致了库尼塔和Carmona/Nualart在20世纪90年代发展的非线性随机整合理论的新应用。此外，我们将开发新的解析和数值方法来解决单边和双边奇异控制问题，建立它们与随机过程的某些表示的关系。目前在数学金融中研究的绝大多数模型将资产价格动态指定为一个外生给定的过程，它独立于市场参与者所采用的交易策略而演变。虽然这种理想化适合于流动性市场(例如，交易国债或蓝筹股)，但在资产价格与交易员产生的需求直接相关的市场中，如何评估流动性风险目前基本上是开放的。开发和分析捕捉价格和交易策略之间的这种非线性反馈影响的数学模型，对本项目所涉及的随机优化和控制提出了挑战。

The project focuses on novel approaches to stochastic optimization problems arising in Mathematical Finance and Economics. A key topic is the development and analysis of new mathematical models for illiquid financial markets which account for the dependence between asset price dynamics and trading strategies, leading to new applications for the nonlinear stochastic integration theory developed by Kunita and Carmona/Nualart in the 1990s. Furthermore, we will develop new analytic and numerical methods to both one- and two-sided singular control problems, building on their relation to certain representations of stochastic processes.The vast majority of models presently studied in Mathematical Finance specify asset price dynamics as an exogenously given process, which evolves independently from the trading strategies employed by the market participants. While this idealization is appropriate for liquid markets (trading, for instance, treasury bonds or blue chips), it is largely open at present how to assess the liquidity risk in markets where asset prices are directly related to the demand generated by traders. The development and analysis of mathematical models capturing this nonlinear feedback affect between prices and trading strategies poses a challenge for stochastic optimization and control that is addressed by this project.