Conditional submodular Choquet expected values and conditional coherent risk measures

We provide an axiomatic definition of conditional submodular capacity that allows conditioning on “null” events and is the basis for the notions of consistency and of consistent extension of a partial assessment. The same definition gives rise to an axiomatic definition of conditional submodular Choquet expected value, which is a conditional functional defined on conditional gambles, that can be expressed as the Choquet integral with respect to its restriction on conditional indicators. Finally, the notion of conditional submodular Choquet expected value is used to provide a definition of conditional submodular coherent risk measure that, locally on every conditioning event, has an upper expected loss interpretation.