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Control Optim."],"published-print":{"date-parts":[[2026,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this paper, we show that the value functions of mean field control problems with common noise are the unique viscosity solutions to fully second-order Hamilton\u2013Jacobi\u2013Bellman equations in a Crandall\u2013Lions-like framework. We allow the second-order derivative in measure to be state-dependent and thus infinite-dimensional, rather than derived from a finite-dimensional operator, and hence we use the term \u201cfully.\u201d Our argument leverages the construction of smooth approximations from particle systems developed by Cosso et al. [ Trans. Amer. Math. Soc., 377 (2024), pp. 31\u201383] and the compactness argument via penalization of measure moments in Soner and Yan [ Appl. Math. Optim., 90 (2024), 1]. Our work addresses unbounded dynamics and state-dependent common noise volatility, and to our knowledge, this is the first result of its kind in the literature.<\/jats:p>","DOI":"10.1137\/24m1717968","type":"journal-article","created":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T07:24:10Z","timestamp":1782890650000},"page":"2250-2280","source":"Crossref","is-referenced-by-count":0,"title":["Viscosity Solutions of Fully Second-Order HJB Equations in the Wasserstein Space"],"prefix":"10.1137","volume":"64","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1926-4570","authenticated-orcid":true,"given":"Erhan","family":"Bayraktar","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hang","family":"Cheung","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ibrahim","family":"Ekren","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8610-6786","authenticated-orcid":true,"given":"Jinniao","family":"Qiu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ho Man","family":"Tai","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, University of Sydney, Camperdown NSW 2050, Australia."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0036-5996","authenticated-orcid":true,"given":"Xin","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Finance and Risk Engineering, New York University, New York, NY 11201 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,7,1]]},"reference":[{"key":"ref1","volume-title":"Gradient Flows: In Metric Spaces and in the Space of Probability Measures","author":"Ambrosio L.","year":"2008","edition":"2"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2018.03.014"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-023-10015-3"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2025.113963"},{"key":"ref5","first-page":"4089","volume":"151","author":"Bayraktar E.","year":"2023","journal-title":"Proc. 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