Convex duality and mathematical finance

Abstract

The theme of this thesis is duality methods in mathematical - nance. This is a hot topic in the eld of mathematical nance, and there is currently a lot of research activity regarding this subject. However, since it is a fairly new eld of study, a lot of the material available is technical and di cult to read. This thesis aims to connect the duality methods used in mathematical nance to the general theory of duality methods in optimization and convexity, and hence clarify the subject. This requires the use of stochastic, real and functional analysis, as well as measure and integration theory. The thesis begins with a presentation of convexity and conju- gate duality theory. Then, this theory is applied to convex risk measures. The nancial market is introduced, and various duality methods, including linear programming duality, Lagrange duality and conjugate duality, are applied to solve utility maximization, pricing and arbitrage problems. This leads to both alternative proofs of known results, as well as some (to my knowledge) new results.