Tools from Convex Analysis, in particular duality methods, arise frequently in Mathematical Finance and Economics. In this course, we will review and motivate some basic concepts in Convex Analysis (Hahn-Banach theorem, biconjugation theorem, subdifferentials, Fenchel-Rockafellar duality theorem) on infinite dimensional spaces.

These tools are going to be applied to a special type of functions which frequently appears in Finance and Economics, in particular when it comes to pricing, risk evaluation or utility measurement. Examples are coherent and convex risk measures, no arbitrage price bounds, good deal bounds and optimized certainty equivalents, among many others. Duality concepts will be discussed and examples including risk measures, portfolio optimization, the Fundamental Theorem of Asset Pricing, hedging and capital/risk allocation problems will be studied.