4 research outputs found
Risk Measures and Upper Probabilities: Coherence and Stratification
Machine learning typically presupposes classical probability theory which
implies that aggregation is built upon expectation. There are now multiple
reasons to motivate looking at richer alternatives to classical probability
theory as a mathematical foundation for machine learning. We systematically
examine a powerful and rich class of alternative aggregation functionals, known
variously as spectral risk measures, Choquet integrals or Lorentz norms. We
present a range of characterization results, and demonstrate what makes this
spectral family so special. In doing so we arrive at a natural stratification
of all coherent risk measures in terms of the upper probabilities that they
induce by exploiting results from the theory of rearrangement invariant Banach
spaces. We empirically demonstrate how this new approach to uncertainty helps
tackling practical machine learning problems
No free lunch and risk measures on Orlicz spaces
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