Cash-Subadditive Risk Measures Without Quasi-Convexity
Bibliographic record
Abstract
In the literature on risk measures, cash subadditivity was proposed to replace cash additivity, motivated by the presence of stochastic or ambiguous interest rates and defaultable contingent claims. Cash subadditivity has been traditionally studied together with quasi-convexity, in a way similar to cash additivity with convexity. In this paper, we study cash-subadditive risk measures without quasi-convexity. One of our major results is that a general cash-subadditive risk measure can be represented as the lower envelope of a family of quasi-convex and cash-subadditive risk measures. Representation results of cash-subadditive risk measures with some additional properties are also examined. The notion of quasi-star-shapedness, which is a natural analogue of star-shapedness, is introduced, and we obtain a corresponding representation result via the lower envelope of normalized, quasi-convex, and cash-subadditive risk measures. Funding: J. Xia received financial support of the National Key R&D Program of China [Grant 2018YFA0703900] and the National Natural Science Foundation of China [Grants 12071146, 12431017, and 12471447]. X. Han, Q. Wang, and R. Wang received financial support from the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2024-03728 and CRC-2022-00141]. X. Han also received financial support from the National Natural Science Foundation of China [Grants 12301604, 12371471, and 12471449]. Q. Wang also received financial support from the Society of Actuaries through the James C. Hickman Scholar Doctoral Stipend.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".