Two-Regime Risk Measures under Convex Loss
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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BibTeX
@misc{nistor2026tworegime,
title = {{Two-Regime Risk Measures under Convex Loss}},
author = {Mihaela-Adriana Nistor and Ionel Popescu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08821},
eprint = {2610.08821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08821},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-Regime Risk Measures under Convex Loss
AU - Mihaela-Adriana Nistor
AU - Ionel Popescu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.08821
UR - https://arxiv.org/abs/2610.08821
AB - 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.
ER -