CAREER: An Approach to Pricing, Hedging, Stability, and Asymptotic Analysis in Financial Markets

随着金融市场不仅发展壮大，而且变得非常复杂，理解它们的定性和定量方面都是数学研究中非常活跃的领域之一。该奖项提供了金融市场的定价、对冲、稳定性和渐近分析方法。这些项目将加深我们对所谓的不完全市场(即抵消风险的能力有限的市场)以及这些市场对不同类型的扰动和交易限制的敏感性的理解。这些主题将导致随机控制、凸性和随机分析的新发展，新的跨学科研究，以及适用于金融行业的结果。该项目包括研究生和博士后研究人员。第一个研究主题是关于扰动下金融市场的稳定性和渐近分析。从数学上讲，这个主题导致研究基本的随机控制问题对输入数据失真的响应。对于扰动和相应的值函数的适当形式的参数化将允许仅涉及基础值函数的部分凸性(在一个变量中)的分析。基本随机控制问题的动态和静态公式都将被考虑。第二个主题是当施加额外的交易限制时，金融工具的定价和对冲。这一主题与研究具有劳动收入的最优投资问题有关，其中额外的交易约束使得问题很难分析，并且需要对劳动收入和价值函数进行特殊形式的参数化。这项数学工作依赖于随机分析、随机控制、有限和无限维凸分析的经典和现代结果，以及首席研究员将建立的新结果。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

As financial markets have not only grown but have also become very complex, understanding both their qualitative and quantitative aspects are among highly active areas of research in mathematics. This award provides methods for pricing, hedging, stability and asymptotic analysis in financial markets. These projects will further our understanding of so-called incomplete markets (i.e., the markets which have a limited capability to offset risks) and the sensitivity of such markets to different types of perturbations and trading restrictions. The topics will lead to new developments in stochastic control, convex and stochastic analysis, to novel interdisciplinary research, and results applicable in the financial industry. Graduate students and post doctoral researchers are included in the work of the project.The first research topic is stability and asymptotic analysis of financial markets with respect to perturbations. Mathematically this topic leads to investigations of the responses of the underlying stochastic control problems to distortions of the input data. An appropriate form of parametrization for the perturbations and the corresponding value functions will allow for analysis involving only partial convexity (in one variable) of the underlying value function. Both dynamic and static formulations of the underlying stochastic control problem will be considered. The second topic is the pricing and hedging of financial instruments when additional trading constraints are imposed. This topic is connected to an investigation of the optimal investment problem with labor income, where extra trading constraints make the problem hard to analyze, and special forms of parametrization of the labor income and the value function are needed. The mathematical work relies on classical and modern results in stochastic analysis, stochastic control, finite and infinite-dimensional convex analysis, and new results to be established by the Principal Investigator.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.