Problems in Mathematical Finance

在这个项目的第一部分，研究者和他的同事们试图使用随机分析的方法来提供金融市场（如股票和固定收益市场）价格形成的全面研究。具体目标是了解特殊风险如何影响均衡价格的形成。目前，很少有可用的结果可以回答这些问题，项目的这一部分寻求提供可用于获得近似答案的可处理模型。在第二部分中，研究者和他的同事寻求开发优化工具，可以处理数学金融中经常遇到的大类随机控制问题。这些问题表现出意想不到的不连续性，使标准数学工具无法适用。从严格的数学角度来看，研究者首先试图在布朗运动控制的连续时间模型中建立不完全平衡的存在。众所周知，这种模型是出了名的难以处理。因此，研究者寻求提供易于处理的近似工具，可以用作一般模型的替代品。其次，研究者和他的同事试图提供一个偏微分方程的最优投资问题的特征与无跨度禀赋。这个控制问题原来有一个不连续的值函数（一个提升或边界层），这阻止了标准偏微分方程技术的使用。最后，研究者和他的同事们试图开发一种工具，可以预先检测到一般类型的随机控制问题的这种不连续性。

LarsenDMS-1411809 In the first part of this project, the investigator and his colleagues seek to use methods from stochastic analysis to provide a comprehensive study of price formation in financial markets such as the stock and fixed-income markets. The specific goal is to understand how idiosyncratic risks affect equilibrium price formation. Currently, there are very few results available that can answer such questions and this part of the project seeks to provide tractable models that can be used to obtain approximate answers. In the second part, the investigator and his colleagues seek to develop optimization tools that can deal with a large class of stochastic control problems often encountered in mathematical finance. These problems exhibit unexpected discontinuities that prevent the standard mathematical tools from being applicable. From a rigorous mathematical perspective, the investigator first seeks to establish the existence of incomplete equilibria in continuous-time models governed by Brownian motions. Such models are well-known for being notoriously intractable. Therefore, the investigator seeks to provide tractable approximation tools that can be used as surrogates for the general models. Secondly, the investigator and his colleagues seek to provide a partial differential equation characterization of the problem of optimal investment with unspanned endowment. This control problem turns out to have a discontinuous value function (a facelift or boundary layer), which prevents the use of standard partial differential equation techniques. Finally, the investigator and his colleagues seek to develop tools that can detect up front such discontinuities for a general class of stochastic control problems.