Problems in mathematical finance

本研究计划有两个项目。 第一个主题涉及“有约束的资产配置”，第二个主题涉及“信用风险模型的综合”。具体而言，第一个主题的目的是从投资者的角度分析稳健的资产配置作用。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，这个问题可以归结为在约束条件下最大化期望效用的问题。我们打算解决以下三个问题：（1）。当约束条件是美式的，如动态VaR约束时，解决问题;解决市场（动态）不完全时的问题，例如当投资者被排除在卖空之外时;（3）。当投资者担心模型设定错误时，解决问题。以上三个问题的解决将加深我们对当前投资领域的理解，并具有实际应用价值。我的第二个项目是关于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，结合这两种方法的最佳功能元素。本项目的目标是扩展结构性信用风险方法。这种结构性信用风险模型的扩展将具有理论上的吸引力和相当大的实用价值。例如，它为开发和实施统一的风险模型开辟了道路，该模型可以适应信贷风险、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.