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.