{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,6]],"date-time":"2026-07-06T22:25:17Z","timestamp":1783376717729,"version":"3.54.6"},"reference-count":53,"publisher":"Institute for Operations Research and the Management Sciences (INFORMS)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics of OR"],"published-print":{"date-parts":[[2026,5]]},"abstract":"<jats:p>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.<\/jats:p>\n                  <jats:p>Funding: J. Xia received financial support of the National Key R&amp;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.<\/jats:p>","DOI":"10.1287\/moor.2022.0312","type":"journal-article","created":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T13:00:13Z","timestamp":1747314013000},"page":"1205-1226","source":"Crossref","is-referenced-by-count":5,"title":["Cash-Subadditive Risk Measures Without Quasi-Convexity"],"prefix":"10.1287","volume":"51","author":[{"given":"Xia","family":"Han","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Nankai University, Tianjin 300071, China; and Key Laboratory of Pure Mathematics and Combinatorics, Nankai University, Tianjin 300071, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9671-8425","authenticated-orcid":false,"given":"Qiuqi","family":"Wang","sequence":"additional","affiliation":[{"name":"Maurice R. Greenberg School of Risk Science, Georgia State University, Atlanta, Georgia 30303"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3849-4555","authenticated-orcid":false,"given":"Ruodu","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6295-5425","authenticated-orcid":false,"given":"Jianming","family":"Xia","sequence":"additional","affiliation":[{"name":"Key Laboratory of Random Complex Structures and Data Science, National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"109","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177704255"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1137\/22M1540090"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9965.00068"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-005-0152-0"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1093\/rfs\/14.2.371"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.1050.0476"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1137\/21M1444278"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1080\/14697688.2017.1297535"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0531(03)00122-4"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1287\/opre.2022.2303"},{"issue":"4","key":"B11","first-page":"743","volume":"21","author":"Cerreia-Vioglio S","year":"2011","journal-title":"Math. Finance"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1093\/rfs\/hhn081"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.5802\/aif.53"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.1524\/strm.2013.1132"},{"key":"B15","unstructured":"Dan\u00edelsson J, Embrechts P, Goodhart C, Keating C, Muennich F, Renault O, Shin H S (2001) An academic response to Basel II. LSE Special Paper Series May 2001, Institution is London School of Economics and Political Science, London."},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04790-3_1"},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1120.0560"},{"key":"B18","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9965.2009.00380.x"},{"key":"B19","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-013-0220-9"},{"key":"B20","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-008-0069-5"},{"key":"B21","doi-asserted-by":"publisher","DOI":"10.1007\/s007800200072"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1515\/9783110463453"},{"key":"B23","doi-asserted-by":"publisher","DOI":"10.1137\/09078033X"},{"key":"B24","doi-asserted-by":"publisher","DOI":"10.1016\/S0378-4266(02)00270-4"},{"key":"B25","doi-asserted-by":"publisher","DOI":"10.1111\/mafi.12028"},{"key":"B26","doi-asserted-by":"publisher","DOI":"10.1016\/j.jet.2003.12.004"},{"key":"B27","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9965.02003"},{"key":"B28","unstructured":"Jia G, Xia J, Zhao R (2020) Monetary risk measures. Preprint, submitted December 12, https:\/\/arxiv.org\/abs\/2012.06751."},{"key":"B29","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2021.12.012"},{"key":"B30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmateco.2019.05.002"},{"key":"B31","doi-asserted-by":"publisher","DOI":"10.1111\/j.1468-0262.2006.00716.x"},{"key":"B32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2014.12.001"},{"key":"B33","doi-asserted-by":"publisher","DOI":"10.1137\/18M121842X"},{"key":"B34","doi-asserted-by":"publisher","DOI":"10.1111\/1468-0262.00303"},{"key":"B35","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2015.0711"},{"key":"B36","volume-title":"Quantitative Risk Management: Concepts, Techniques and Tools","author":"McNeil AJ","year":"2015"},{"key":"B37","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2021.109345"},{"key":"B38","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.2016.2486"},{"key":"B39","unstructured":"Munari CA (2015) Measuring risk beyond the cash-additive paradigm. Doctoral dissertation, ETH Zurich, Zurich."},{"key":"B40","doi-asserted-by":"publisher","DOI":"10.3982\/TE5799"},{"key":"B41","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2681(82)90008-7"},{"key":"B42","doi-asserted-by":"publisher","DOI":"10.1016\/j.insmatheco.2024.05.002"},{"key":"B43","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-33590-7"},{"key":"B44","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1986-0835875-8"},{"key":"B45","doi-asserted-by":"publisher","DOI":"10.2307\/1911053"},{"key":"B46","doi-asserted-by":"publisher","DOI":"10.1007\/s11425-006-2010-8"},{"key":"B47","doi-asserted-by":"publisher","DOI":"10.1016\/j.insmatheco.2009.09.011"},{"key":"B48","doi-asserted-by":"publisher","DOI":"10.1007\/s11579-014-0136-y"},{"key":"B49","doi-asserted-by":"publisher","DOI":"10.1002\/9781118468333.ch3"},{"key":"B50","doi-asserted-by":"crossref","unstructured":"Wang R, Wu Q (2020) Dependence and risk attitudes: An equivalence. Preprint, submitted November 23, https:\/\/dx.doi.org\/10.2139\/ssrn.3707709.","DOI":"10.2139\/ssrn.3707709"},{"key":"B51","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.2020.3617"},{"key":"B52","doi-asserted-by":"crossref","unstructured":"Xia J (2013) Comonotonic convex preferences. Preprint, submitted July 27, https:\/\/dx.doi.org\/10.2139\/ssrn.2298884.","DOI":"10.2139\/ssrn.2298884"},{"key":"B53","doi-asserted-by":"publisher","DOI":"10.2307\/1911158"}],"container-title":["Mathematics of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubsonline.informs.org\/doi\/pdf\/10.1287\/moor.2022.0312","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T08:25:24Z","timestamp":1778228724000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubsonline.informs.org\/doi\/10.1287\/moor.2022.0312"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5]]},"references-count":53,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["10.1287\/moor.2022.0312"],"URL":"https:\/\/doi.org\/10.1287\/moor.2022.0312","relation":{},"ISSN":["0364-765X","1526-5471"],"issn-type":[{"value":"0364-765X","type":"print"},{"value":"1526-5471","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5]]}}}