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Dual representation for star-shaped MARRMs via transfer of techniques from star-shaped monetary risk measures

Determine whether dual representation techniques developed for star-shaped monetary risk measures can be transferred to star-shaped multi-asset return risk measures, and construct a corresponding minmax-type dual representation for star-shaped MARRMs.

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Background

The paper provides dual representations for logconvex and quasi-logconvex MARRMs and a minmax-type representation for log-star-shaped cases via the log-domain connection. However, for star-shaped MARRMs defined directly on C, it is unknown whether the duality frameworks for star-shaped monetary risk measures (as in Castagnoli et al. and related work) can be adapted.

The authors highlight that a successful transfer would likely yield a minmax-type dual representation analogous to the star-shaped case treated earlier, but they leave this as an open problem.

References

Regarding these difficulties, we leave the following points open for future research: (2) If the MARRM is not convex, but star-shaped, is it possible to transfer the ideas to obtain dual representations for star-shaped monetary risk measures (see Castagnoli, Laeven2) to the case of MARRMs? Note that the latter would result in a "minmax" representation as in Theorem 4.??.

— Multi-asset return risk measures (2411.08763 - LaudagĂ© et al., 13 Nov 2024) in Subsection "Duality without Assumption" in Section "Dual Representations of MARRM"