Financial Mathematics: Nonlinear Problems and Systemic Risk

在第一个项目中，提出建立非线性偏偏微分方程的渐近分析和均匀化理论的新结果，并直接应用于金融业从业者面临的几个重要和实际问题。这些问题包括不确定波动率下的风险管理、短期隐含波动率的行为以及随机波动率下的投资组合优化。在第二个项目中，建议开发基于相互作用扩散系统的新模型，其中耦合通过贷款偏好在漂移中建模。结合复杂的蒙特卡罗方法，包括相互作用粒子系统方法，这些模型将允许研究系统的稳定性和与网络系统风险相关的各种统计量。在第一个项目中解决了在不确定或随机波动波动下风险管理中自然产生的具有挑战性的非线性问题。本研究对金融行业从业者面临的实际问题具有直接的应用价值。第二个项目是研究系统风险和银行系统稳定性（或不稳定性）的数学分析的新方向。最近的金融危机表明，人们对银行网络中一连串违约的风险缺乏了解。建议开发新的模型，以研究系统的稳定性和与网络系统风险相关的各种统计量。这项研究，包括其培训部分，预计将有助于监管机构最近成立的金融研究办公室（Office of Financial research）所开展的工作。

In the first project, it is proposed to establish new results in the theory of asymptotic analysis and homogenization of nonlinear partial partial differential equations with direct applications to several important and practical problems faced by practitioners in the financial industry. These problems include risk management under uncertain volatility, behavior of implied volatilities at short maturities, and portfolio optimization under stochastic volatility. In the second project, it is proposed to develop new models based on systems of interacting diffusions where the coupling is modeled in the drifts through lending preferences. Combined with sophisticated Monte Carlo methods, including interacting particle system methods, these models will allow to study the stability of the system and the various statistical quantities relevant to systemic risk of a network.Challenging nonlinear problems arising naturally in the context of risk management under uncertain or randomly fluctuating volatility are addressed in the first project. This research has direct applications to practical problems faced by practitioners in the financial industry. The second project is a new direction of research on systemic risk and mathematical analysis of the stability (or instability) of our banking system. The recent financial crisis has revealed a lack of understanding of the risk of cascade of defaults in banking networks. It is proposed to develop new models which will allow to study the stability of the system and the various statistical quantities relevant to systemic risk of a network. This research, including its training component, is expected to contribute to the effort started by the regulators in the recent creation of the Office of Financial Research.