Dual representation of MARRMs without passing to the log domain

Derive a dual representation for the multi-asset return risk measure η_R(X) = inf{ π(Z) : Z ∈ S, X/Z ∈ B } directly on the multiplicative domain C ⊂ L^0_{++}, without relying on the log-transformation to the additive domain C_log and the associated multi-asset risk measure framework.

Background

Throughout Section "Dual Representations of MARRM," the paper obtains dual representations by rewriting a multi-asset return risk measure (MARRM) as the exponential of a multi-asset risk measure (MARM) applied to log-returns, which requires working on the log-domain C_log and imposing local convexity (Assumption on C_log).

The authors note that typical domains arising from return risk measures (e.g., L^∞_{++}) yield C_log that is only a convex cone and not a linear space, so the duality results in the paper apply only to restricted domains. They explicitly state that obtaining dual representations starting from C, without passing to C_log, remains open.