AMC-SS: Problems in Mathematical Finance

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。量化，管理和定价风险不仅对经济中的大玩家很重要，而且对每个人都变得越来越重要，他们试图通过退休基金来减少他们的财富所带来的风险，以及代表他们的机构投资者。金融市场为经济的其他部分提供了必要的流动性，使其能够良好运作，而不了解所承担风险的性质，错误定价风险和监管不足，可能会对我们社会的福利产生巨大影响，最近的危机再次证明了这一点。因此，人们在金融数学领域进行了大量的研究工作，以解决新的数学挑战。这个新领域的根源在于随机分析，也经常与其他数学领域相互作用，如偏微分方程理论和泛函分析。 这个项目解决了一些关于该领域的重要模型的基本问题，这些问题仍然没有答案。 此外，还将开发考虑到观测的离散、异步和非平稳性质的新模型，并将解决新问题的数学和计算挑战。从退休人员的角度（与客观标准）和机构投资者的角度（与监管机构施加的风险约束）的优化问题进行了分析。本课题的另一个目标是，为发行人的违约（关联）提供保险的结构性信用产品，开发简约模型并分析其逆问题。具体而言，本课题的第一部分将解决最优停止理论中的开放性问题和金融数学中的自由边界问题。在第二部分的解决方案有许多实际的控制问题的直接应用，因为在许多应用中的观察的性质是离散的，异步的和非平稳的（即观察源的性质随时间变化）。在第三部分中，将考虑具有终生破产概率、占用时间或其他优化和/或风险约束的效用最大化问题。将讨论具有非线性期望的最优控制/停止问题。这些导致新的数学挑战，这将导致新的方法的发展。研究不确定投资环境和学习对最优投资策略的影响。 去年信用衍生品市场的巨大损失表明，信用衍生品的适当建模和定价是一个具有挑战性的开放性问题。在第四条路径中，研究人员将开发有效的模型，解决相应的逆问题并为场外期权定价。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Quantifying, managing and pricing risk is not only important for the big players in the economy but has become increasingly important for every individual, who are trying to reduce the risk of outliving their wealth through their retirement funds, and the institutional investors representing them. The financial markets provide the necessary liquidity for the rest of the economy to function well, and not understanding the nature of the risk taken, the mispricing of the risk and an inadequate oversight, can have dramatic impacts on the welfare of our society, as the latest crisis demonstrated one more time. Consequently there has been an intense research effort, which is embodied in the field of financial mathematics, to solve the new mathematical challenges. This new field takes its roots in stochastic analysis and also frequently interacts with other fields of mathematics such as the theory of partial differential equations and functional analysis. This project resolves some fundamental questions concerning the important models of the field that remain unanswered. Additionally, new models which take into account the discrete, asynchronous and non-stationary nature of the observations will be developed and the mathematical and computational challenges for new problems of interest will be resolved. Optimization problems from a retiree's perspective (with an objective criterion) and an institutional investor perspective (with risk constraints imposed by regulators) are to be analyzed. Developing parsimonious models and analyzing the inverse problems for structured credit products, which were developed to provide insurance against (correlated) default of issuers, is another goal.More specifically, the first part of the proposed project will settle open problems in the theory of Optimal Stopping and related free boundary problems that arise in Financial Mathematics. The solutions in the second part have direct applications to many practical control problems since the nature of observations in many applications is discrete, asynchronous and non-stationary (i.e. the nature of the source of observation change with time). In the third part utility maximization problems with probability of lifetime ruin, occupation time, or other optimization and/or risk constraints will be considered. Optimal control/stopping problems with non-linear expectations will be discussed. These lead to new mathematical challenges which will lead to new methodological developments. The effects of uncertain investment environment and learning on the optimal investment strategies will also be investigated. The dramatic losses in the credit derivative markets last year shows that appropriate modeling and pricing of credit derivatives is a challenging open problem. In the fourth path, the investigator will develop effective models, solve the corresponding inverse problems and price over the counter options.