Dual representations for quasiconvex compositions with applications to systemic risk measures
Çağın Ararat, Mücahit Aygün
TL;DR
The paper addresses dual representations for quasiconvex systemic risk measures by treating systemic risk as a quasiconvex composition R=ρ∘Λ and developing explicit penalty-function formulas α_{f∘g} in terms of the penalties of the ingredients. A nonstandard minimax approach yields the main theorem linking α_{f∘g} to α_f and α_{y^*∘g}, enabling decomposition of contributions from the risk measure and aggregation function in probabilistic settings. The authors apply the theory to concrete examples, including certainty- equivalents and the Eisenberg–Noe clearing model, providing detailed economic interpretations of the dual variables and their associated divergences. The results offer a flexible framework for quasiconvex systemic risk estimation in infinite-dimensional spaces and illustrate how network structure and model uncertainty influence dual representations and risk assessment. Overall, the work extends convex duality to the quasiconvex regime, with significant implications for systemic risk measurement and risk management in interconnected financial networks.
Abstract
Motivated by the problem of finding dual representations for quasiconvex systemic risk measures in financial mathematics, we study quasiconvex compositions in an abstract infinite-dimensional setting. We calculate an explicit formula for the penalty function of the composition in terms of the penalty functions of the ingredient functions. The proof makes use of a nonstandard minimax inequality (rather than equality as in the standard case) that is available in the literature. In the second part of the paper, we apply our results in concrete probabilistic settings for systemic risk measures, in particular, in the context of Eisenberg-Noe clearing model. We also provide novel economic interpretations of the dual representations in these settings.