Some Topics in Robust Mathematical Finance

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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.