Some Topics in Robust Mathematical Finance

模型不确定性在金融和经济学中普遍存在，近年来受到越来越多的关注。本文建议采用稳健方法研究不确定条件下数学金融学的两个基本问题--定价和风险度量，这是目前数学金融学的一个主要研究趋势。 1.稳健定价理论。其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，这两个定理是经典数学定价理论的基石。尽管各种研究者对这一领域进行了大量的研究，但一个相对完整的理论仍有待建立。 目标： -在稳健的环境下建立FTAP和ST的一般版本； -系统地研究了容度函数空间。 2.不确定性环境下的风险度量。目标是研究不确定性如何影响风险度量，以及如何在风险度量中考虑不确定性。 目标： -研究容限函数空间上的风险度量； -确定受相关性约束的相关性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.