Some Topics in Robust Mathematical Finance

模型不确定性是金融经济学中普遍存在的问题，近年来受到越来越多的关注。我提出在不确定性条件下，采用稳健方法研究数学金融的两个基础课题——定价和风险计量，稳健方法是目前数学金融研究的一个主要趋势。******1。稳健定价理论。目标是建立资产定价基本定理（FTAP）和超级对冲定理（ST）的稳健版本，它们是经典数学定价理论的基石。尽管各路研究者对这一领域进行了大量的研究，但一个较为全面的理论还有待建立。******目标：***-建立一个通用版本的FTAP和ST在稳健设置；***-系统地研究容量函数空间。******不确定性下的风险度量。目的是研究不确定性对风险度量的影响，以及在风险度量中如何考虑不确定性。******目标：***-研究容量功能空间的风险措施；***-在相关性约束下确定依赖不确定性下的风险聚集边界。***********

Model uncertainty is pervasive in finance and economics, and has been gaining more and more attention in recent years. I propose to study the two fundamental subjects in mathematical finance, Pricing and Risk Measurement, under uncertainty, using the robust approach, which is currently a leading research trend in mathematical finance.******1. Robust pricing theory. The goal is to establish robust versions of the Fundamental Theorem of Asset Pricing (FTAP) and the Superhedging Theorem (ST), which are cornerstones of the classical mathematical pricing theory. Although many efforts have been devoted to this area by various researchers, a relatively comprehensive theory is still to be established.******Goals:***- establish a general version of FTAP and ST in the robust setting;***- systematically investigate capacitary function spaces.******2. Risk measurement under uncertainty. The goal is to study how risk measurement is affected by uncertainty and how to take uncertainty into account in risk measurement. ******Goals:***- study risk measures on capacitary function spaces;***- determine risk aggregation bounds under dependence uncertainty subject to correlation constraint.***********