Dual representations for quasiconvex compositions with applications to systemic risk measures

TL;DR

This paper develops explicit dual representations for quasiconvex systemic risk measures in infinite-dimensional spaces, with applications to financial models like Eisenberg-Noe, offering new economic insights.

Contribution

It derives an explicit formula for the penalty function of quasiconvex compositions and applies it to systemic risk measures in probabilistic financial models.

Findings

Explicit penalty function formula for quasiconvex compositions

Application to Eisenberg-Noe systemic risk model

New economic interpretations of dual representations

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.