Two-Regime Risk Measures under Convex Loss

Mihaela-Adriana Nistor; Ionel Popescu

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arXiv · 2610.08821

Two-Regime Risk Measures under Convex Loss

Mihaela-Adriana Nistor·Ionel Popescu

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Abstract

We study a two-regime summary of a real-valued loss distribution. The two representative levels and the boundary between them are chosen by minimizing a convex residual loss. When the distribution has an atom at the boundary, assigning that atom to the lower or upper regime can give different optimized costs. Taking the better whole-atom assignment yields the lower-semicontinuous cutoff profile. We prove epi-convergence of this profile under a loss-adapted \(ψ\)-weak topology and obtain convergence of minimum values and outer stability of optimal cutoff sets. Under a unique finite atom-free limiting cutoff, positive regime masses, and unique conditional centers, the two fitted levels and a canonical one-jump representation converge as well. We also examine the resulting summary from the viewpoint of monetary risk axioms, establish explicit failures of external monotonicity and subadditivity, and give finite-support algorithms with conditional error control. A four-scenario model illustrates the cutoff geometry and the role of ties and atoms.

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Mihaela-Adriana Nistor, Ionel Popescu. 2026-09-24. Two-Regime Risk Measures under Convex Loss. https://arxiv.org/abs/2610.08821

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