A Study on Dilatation Monotone Maps and the Haezendonck-Goovaerts Risk Measure

Risk measures have been studied for several decades in the financial literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some space X to the (extended) real line. It is difficult to give a reliable assessment of financial risk without a suitable model space X where the financial position lies there. 

In this work, we assume the domain X for the risk measure ρ : X → (−∈,∈] is a subset of L¹ that contains the space of simple random variables L and has the Fatou property. We focus on dilatation monotonicity as desirable preferences of the decision-maker over risks. We discuss continuity properties of dilatation monotone risk measures on X. We prove that on a non-atomic probability space, every dilatation monotone convex risk measure on X can extend uniquely to a σ(L¹,L)-lower semicontinuous dilatation monotone convex risk measure. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. 

In the second part of this thesis, we overview various properties of the Haezendonck–Goovaerts risk measure as a dilatation monotone risk measure. In the special case Φ(x) = x², we develop an explicit expression for the Haezendonck–Goovaerts risk measure when the risk variable X follows some specific distributions and propose an empirical algorithm to estimate this risk measure. Moreover, we investigate the performance of some members of this class of risk measures on real data as a possible application of the Haezendonck-Goovaerts risk measure.