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。我建议使用稳健方法研究数理金融学中的两个基本主题，定价和风险度量，在不确定性下，这是目前数理金融学的主要研究趋势。稳健定价理论目标是建立资产定价基本定理（FTAP）和超对冲定理（ST）的稳健版本，这是经典数学定价理论的基石。虽然许多努力已经致力于这一领域的各种研究人员，一个相对全面的理论仍然有待established.Goals：-建立一个一般版本的FTAP和ST在强大的设置;-系统地调查容量功能空间。不确定性下的风险度量。研究不确定性对风险度量的影响以及如何在风险度量中考虑不确定性。目标：-研究容性函数空间上的风险测度;-确定相关约束下依赖不确定性下的风险聚集界。

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.