We study risk measures for financial positions in a multi-asset setting,
representing the minimum amount of capital to raise and invest in eligible
portfolios of traded assets in order to meet a prescribed acceptability
constraint. We investigate finiteness and continuity properties of these
multi-asset risk measures, highlighting the interplay between the acceptance
set and the class of eligible portfolios. We develop a new approach to dual
representations of convex multi-asset risk measures which relies on a
characterization of the structure of closed convex acceptance sets. To avoid
degenerate cases we need to ensure the existence of extensions of the
underlying pricing functional which belong to the effective domain of the
support function of the chosen acceptance set. We provide a characterization of
when such extensions exist. Finally, we discuss applications to conical market
models and set-valued risk measures, optimal risk sharing, and super-hedging
with shortfall risk.
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