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

该项目侧重于数学金融和经济学中出现的随机优化问题的新方法。 一个关键的主题是开发和分析新的数学模型，为非流动性的金融市场，占资产价格动态和交易策略之间的依赖关系，导致新的应用程序的非线性随机集成理论开发的Kunita和Carmona/Nualart在20世纪90年代。 此外，我们将开发新的分析和数值方法，以单边和双边的奇异控制问题，建立在它们的关系，以某些表示的随机processes.The目前研究的数学金融模型的绝大多数指定资产价格动态作为一个外生给定的过程，独立于市场参与者所采用的交易策略。 虽然这种理想化适用于流动性市场（例如，国债或蓝筹股的交易），但在资产价格与交易者产生的需求直接相关的市场中，如何评估流动性风险目前在很大程度上还没有定论。 数学模型的开发和分析捕捉价格和交易策略之间的非线性反馈影响，这对本项目所解决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.