Problems in mathematical finance

这个研究计划有两个项目。第一部分是“约束下的资产配置”，第二部分是“信用风险模型的综合”。具体而言，第一个主题的目标是从投资者的角度分析稳健的资产配置角色。假设投资者有风险偏好和其他特定目标或限制。从数学上讲，这个问题可以简化为在约束条件下最大化预期效用的问题。我们打算解决以下三个问题：(1)。解决美式约束条件下的问题，如动态VaR约束；（2）. 解决市场（动态）不完整时的问题，例如当投资者被禁止卖空时；（3). 解决投资者担心模型规格错误的问题。以上三个问题的解决将加深我们目前在投资领域的认识，并具有实际应用价值。我的第二个项目涉及信用风险模型的综合。目前有两种相关的方法：结构形式和简化形式在某些情况下都是成功的。然而，该领域缺乏一个统一的框架来结合这两种方法的最佳特性元素。该项目的目的是推广结构性信用风险方法。这种对结构性信用风险模型的拓展，既有理论号召力，又有相当的实用价值。例如，它为开发和实施统一的风险模型开辟了道路，该模型可以容纳信用风险、股权风险和利率风险。

There are two projects in this research proposal.  The first one deals with ``Asset Allocation with constraints" and the second one concerns ``A Synthesis of Credit Risk Models".Specifically, the objective of the first topic is to analyze robust asset allocation roles from the investor's perspective. Investors are assumed to have risk preferences and other specific objectives or restrictions. Mathematically speaking, the problem can be reduced to one of maximizing expected utility subject to constraints. We intend to tackle the following three questions: (1). Solve the problem when the constraints are American-style, for instance, the dynamic VaR constraint; (2). Solve the problem when the market is (dynamically) incomplete, for instance when investors are precluded from short selling; (3). Solve the problem when investors have concerns about model misspecification. The solutions of the above three questions would deepen our current understanding in the investment area and also have practical applications.My second project concerns a synthesis of credit risk models. Currently there are two relevant approaches: Both the structural and the reduced-form are successful in certain circumstances. However the field lacks a uniform framework that combines the best features elements of these two approaches. The objective of this project is to extend the structural credit risk approach. This extension of the structural credit risk model will have both theoretical appeal and considerable practical value. For instance, it opens the way to develop and implement a uniform risk model that can accommodate credit risk, equity risk and interest rate risk.