Deriving the modified VaR elicitation results directly from Rostek’s axiomatization

Ascertain whether the calibration results for Value-at-Risk capacities—specifically, Proposition 5.1 (for γ ≤ 1/2, defining the set function v via the recursion g_0(A) = right-Value-at-Risk at level 1−γ applied to 1_A; g_t(A) = sup over B in Σ of the infimum over C ⊆ A^c of g_{t−1}(A ∪ B) + g_{t−1}(A ∪ C) − g_{t−1}(B ∪ C), truncated at 1; and showing inf LC_v = (1/γ) P) and Proposition 5.2 (for γ > 1/2, defining the set function w via the recursion h_0(A) = Value-at-Risk at level γ applied to 1_A; h_t(A) defined analogously; and showing sup LA_w = (1/(1−γ)) P)—can be obtained directly or more simply from Rostek (2010)’s axiomatic characterization of quantile preferences, rather than through the bespoke recursive construction presented here. Determine whether such a derivation exists and, if it does, provide a streamlined proof that recovers the reference probability measure P and the quantile level γ from the axioms and representation results of Rostek (2010).

Background

The paper develops a modified elicitation procedure to recover the reference probability measure P and the quantile level γ for Value-at-Risk (VaR) capacities when standard supporting-set methods yield only trivial extrema. The method constructs auxiliary set functions (v for γ ≤ 1/2 via the right VaR and w for γ > 1/2 via the left VaR) using recursive sup–inf operations; these yield inf LC_v = (1/γ) P and sup LA_w = (1/(1−γ)) P, thereby recovering both P and γ.

Rostek (2010) provides an axiomatic characterization of quantile preferences in a Savage-style framework. The authors note that their VaR calibration results (Propositions 5.1 and 5.2) may plausibly be linked to Rostek’s representation, but they explicitly state uncertainty about whether—and how—those results could be derived more simply from Rostek’s axioms and representation theorems. This raises a methodological open question about connecting their constructive elicitation approach to existing axiomatic frameworks.