Problems in mathematical finance

点击翻译按钮获取中文摘要

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.