No free lunch and risk measures on Orlicz spaces

Abstract

The importance of Orlicz spaces in the study of mathematics of nance came to the for in the 2000's when Frittelli and his collaborators connected the theory of utility functions to Orlicz spaces. In this thesis, we look at how Orlicz spaces play a role in nancial mathematics. After giving an overview of scalar-valued Orlicz spaces, we look at the rst fundamental theorem of asset pricing in an Orlicz space setting. We then give a brief summary of scalar risk measures, followed by the representation result for convex risk measures on Orlicz hearts. As an example of a risk measure, we take a detailed look at the Wang transform both as a pricing mechanism and as a risk measure. As the theory of nancial mathematics is moving towards the set-valued setting, we give a description of vector-valued Orlicz hearts and their duals using tensor products. Lastly, we look at set-valued risk measures on Orlicz hearts, proving a robust representation theorem via a tensor product approach.