On Quantile Risk Measures and Their Domain

In the present paper we study quantile risk measures and their domain. Our starting point is that, for a probability measure $Q$ on the open unit interval and a wide class $\mathcal{L}_Q$ of random variables, we define the quantile risk measure $\varrho_Q$ as the map which integrates the quantile function of a random variable in $\mathcal{L}_Q$ with respect to $Q$. The definition of $\mathcal{L}_Q$ ensures that $\varrho_Q$ cannot attain the value $+\infty$ and cannot be extended beyond $\mathcal{L}_Q$ without losing this property. The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view at the distortion risk measures generated by a distribution function on the unit interval. In this general setting, we prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall.