A framework for measures of risk under uncertainty

A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also on various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantify jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalised risk measure. We present a series of relevant theoretical results. The worst-case, coherent and robust generalised risk measures are characterised via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law-invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law-invariance are derived from a combination of other conditions, which provides additional support for coherent risk measures such as expected shortfall over value-at-risk, a relevant issue for risk management practice.