{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T23:18:05Z","timestamp":1783207085355,"version":"3.54.6"},"reference-count":45,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62561160159"],"award-info":[{"award-number":["62561160159"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12521001"],"award-info":[{"award-number":["12521001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11831010"],"award-info":[{"award-number":["11831010"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2023YFA1009200"],"award-info":[{"award-number":["2023YFA1009200"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Systems & Control Letters"],"published-print":{"date-parts":[[2026,9]]},"DOI":"10.1016\/j.sysconle.2026.106514","type":"journal-article","created":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T13:26:23Z","timestamp":1782480383000},"page":"106514","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["The stochastic maximum principle for mean-field relaxed control problem with regime-switching"],"prefix":"10.1016","volume":"216","author":[{"given":"Jincan","family":"Song","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0758-9463","authenticated-orcid":false,"given":"Zhen","family":"Wu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.sysconle.2026.106514_b1","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/0022-247X(64)90070-8","article-title":"A maximum principle for stochastic control systems","volume":"8","author":"Kushner","year":"1964","journal-title":"J. Math. Anal. Appl."},{"key":"10.1016\/j.sysconle.2026.106514_b2","series-title":"Nonlinear Filtering and Stochastic Control (Cortona, 1981)","first-page":"1","author":"Bensoussan","year":"1982"},{"key":"10.1016\/j.sysconle.2026.106514_b3","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/BF01447329","article-title":"The optimal control of diffusions","volume":"22","author":"Elliott","year":"1990","journal-title":"Appl. Math. Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b4","doi-asserted-by":"crossref","first-page":"966","DOI":"10.1137\/0328054","article-title":"A general stochastic maximum principle for optimal control problems","volume":"28","author":"Peng","year":"1990","journal-title":"SIAM J. Control Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b5","unstructured":"R. Situ, A maximum principle for optimal controls of stochastic systems with random jumps, in: Proc. of the National Conference on Control Theory and Applications, 1991."},{"key":"10.1016\/j.sysconle.2026.106514_b6","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1137\/S0363012992233858","article-title":"Necessary conditions for optimal control of stochastic systems with random jumps","volume":"32","author":"Tang","year":"1994","journal-title":"SIAM J. Control Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b7","first-page":"249","article-title":"Maximum principle for optimal control problem of fully coupled forward\u2013backward stochastic systems","volume":"11","author":"Wu","year":"1998","journal-title":"Syst. Sci. Math. Sci."},{"key":"10.1016\/j.sysconle.2026.106514_b8","first-page":"161","article-title":"The maximum principle for fully coupled forward\u2013backward stochastic control system","volume":"32","author":"Shi","year":"2006","journal-title":"Acta Automat. Sinica"},{"key":"10.1016\/j.sysconle.2026.106514_b9","doi-asserted-by":"crossref","first-page":"2171","DOI":"10.1137\/19M1292308","article-title":"The maximum principle for progressive optimal stochastic control problems with random jumps","volume":"58","author":"Song","year":"2020","journal-title":"SIAM J. Control Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b10","doi-asserted-by":"crossref","first-page":"664","DOI":"10.3934\/mcrf.2022012","article-title":"A general maximum principle for partially observed mean-field stochastic system with random jumps in progressive structure","volume":"13","author":"Chen","year":"2023","journal-title":"Math. Control Relat. Fields"},{"key":"10.1016\/j.sysconle.2026.106514_b11","doi-asserted-by":"crossref","first-page":"221","DOI":"10.4310\/CIS.2006.v6.n3.a5","article-title":"Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle","volume":"6","author":"Huang","year":"2006","journal-title":"Commun. Inf. Syst."},{"key":"10.1016\/j.sysconle.2026.106514_b12","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s11537-007-0657-8","article-title":"Mean field games","volume":"2","author":"Lasry","year":"2007","journal-title":"Jpn. J. Math."},{"key":"10.1016\/j.sysconle.2026.106514_b13","doi-asserted-by":"crossref","first-page":"2809","DOI":"10.1137\/120892477","article-title":"Linear-quadratic optimal control problems for mean-field stochastic differential equations","volume":"51","author":"Yong","year":"2013","journal-title":"SIAM J. Control Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b14","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/s00245-010-9123-8","article-title":"A maximum principle for SDEs of mean-field type","volume":"63","author":"Andersson","year":"2011","journal-title":"Appl. Math. Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b15","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/s00245-011-9136-y","article-title":"A general stochastic maximum principle for SDEs of mean-field type","volume":"64","author":"Buckdahn","year":"2011","journal-title":"Appl. Math. Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b16","first-page":"3133","article-title":"Mean-field backward stochastic differential equations and related partial differential equations, stochastic process","volume":"119","author":"Buckdahn","year":"2009","journal-title":"Appl."},{"key":"10.1016\/j.sysconle.2026.106514_b17","doi-asserted-by":"crossref","first-page":"1736","DOI":"10.1109\/TAC.2012.2183439","article-title":"Social optima in mean field LQG control: centralized and decentralized strategies","volume":"57","author":"Huang","year":"2012","journal-title":"IEEE Trans. Automat. Control"},{"key":"10.1016\/j.sysconle.2026.106514_b18","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1109\/TAC.2012.2215399","article-title":"Nash, social and centralized solutions to consensus problems via mean field control theory","volume":"58","author":"Nourian","year":"2013","journal-title":"IEEE Trans. Automat. Control"},{"key":"10.1016\/j.sysconle.2026.106514_b19","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1137\/S0363012999356325","article-title":"Stock trading: an optimal selling rule","volume":"40","author":"Zhang","year":"2001","journal-title":"SIAM J. Control Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b20","doi-asserted-by":"crossref","first-page":"780","DOI":"10.1137\/090770552","article-title":"Trend following trading under a regime switching model","volume":"1","author":"Dai","year":"2010","journal-title":"SIAM J. Financial Math."},{"key":"10.1016\/j.sysconle.2026.106514_b21","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s00245-010-9130-9","article-title":"Sufficient stochastic maximum principle in a regime-switching diffusion model","volume":"64","author":"Donnelly","year":"2011","journal-title":"Appl. Math. Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b22","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1016\/j.sysconle.2012.06.006","article-title":"Maximum principle for optimal control problems of forward\u2013backward regime-switching system and applications","volume":"61","author":"Tao","year":"2012","journal-title":"Systems Control Lett."},{"key":"10.1016\/j.sysconle.2026.106514_b23","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s13235-019-00306-2","article-title":"Mean-field-type games with jump and regime switching","volume":"10","author":"Bensoussan","year":"2020","journal-title":"Dyn. Games Appl."},{"key":"10.1016\/j.sysconle.2026.106514_b24","doi-asserted-by":"crossref","first-page":"3255","DOI":"10.1007\/s00245-021-09747-x","article-title":"A general stochastic maximum principle for mean-field controls with regime switching","volume":"84","author":"Nguyen","year":"2021","journal-title":"Appl. Math. Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b25","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1137\/22M1481415","article-title":"Stochastic linear-quadratic optimal control problems with random coefficients and Markovian regime switching system","volume":"61","author":"Wen","year":"2023","journal-title":"SIAM J. Control Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b26","series-title":"Lectures on the Calculus of Variations and Optimal Control Theory","author":"Young","year":"1969"},{"key":"10.1016\/j.sysconle.2026.106514_b27","series-title":"Differential Games and Control Theory, II (Proc. 2nd Conf., Univ. Rhode Island, Kingston, R. I., 1976)","author":"Fleming","year":"1977"},{"key":"10.1016\/j.sysconle.2026.106514_b28","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1080\/17442508708833443","article-title":"Compactification methods in the control of degenerate diffusions: existence of an optimal control","volume":"20","author":"Karoui","year":"1987","journal-title":"Stochastics"},{"key":"10.1016\/j.sysconle.2026.106514_b29","first-page":"72762","article-title":"Approximation and optimality necessary conditions in relaxed stochastic control problems","author":"Bahlali","year":"2006","journal-title":"J. Appl. Math. Stoch. Anal."},{"key":"10.1016\/j.sysconle.2026.106514_b30","doi-asserted-by":"crossref","first-page":"2078","DOI":"10.1137\/070681053","article-title":"Necessary and sufficient optimality conditions for relaxed and strict control problems","volume":"47","author":"Bahlali","year":"2008","journal-title":"SIAM J. Control Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.sysconle.2016.12.009","article-title":"Existence and optimality conditions for relaxed mean-field stochastic control problems","volume":"102","author":"Bahlali","year":"2017","journal-title":"Systems Control Lett."},{"key":"10.1016\/j.sysconle.2026.106514_b32","doi-asserted-by":"crossref","DOI":"10.1142\/S0219493718500247","article-title":"On the relaxed mean-field stochastic control problem","volume":"18","author":"Bahlali","year":"2018","journal-title":"Stoch. Dyn."},{"key":"10.1016\/j.sysconle.2026.106514_b33","doi-asserted-by":"crossref","DOI":"10.1016\/j.sysconle.2022.105391","article-title":"The stochastic maximum principle for relaxed control problem with regime-switching","volume":"169","author":"Chen","year":"2022","journal-title":"Systems Control Lett."},{"key":"10.1016\/j.sysconle.2026.106514_b34","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1137\/S0363012996302664","article-title":"Stochastic near-optimal controls: necessary and sufficient conditions for near-optimality","volume":"36","author":"Zhou","year":"1998","journal-title":"SIAM J. Control Optim."},{"key":"10.1016\/j.sysconle.2026.106514_b35","article-title":"Reinforcement learning in continuous time and space: a stochastic control approach","volume":"21","author":"Wang","year":"2020","journal-title":"J. Mach. Learn. Res."},{"key":"10.1016\/j.sysconle.2026.106514_b36","doi-asserted-by":"crossref","DOI":"10.1142\/S0219024925500141","article-title":"Exploratory mean\u2013variance portfolio optimization with regime-switching market dynamics","volume":"28","author":"Chen","year":"2025","journal-title":"Int. J. Theor. Appl. Finance"},{"key":"10.1016\/j.sysconle.2026.106514_b37","first-page":"24","article-title":"On the maximum principle for relaxed control problems of nonlinear stochastic systems","author":"Mezerdi","year":"2024","journal-title":"Adv. Contin. Discret. Model."},{"key":"10.1016\/j.sysconle.2026.106514_b38","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/BF01288560","article-title":"Martingale measures and stochastic calculus","volume":"84","author":"Karoui","year":"1990","journal-title":"Probab. Theory Relat. Fields"},{"key":"10.1016\/j.sysconle.2026.106514_b39","first-page":"1","article-title":"Nonlinear SDEs driven by L\u00e9vy processes and related PDEs","volume":"4","author":"Jourdain","year":"2008","journal-title":"ALEA Lat. Am. J. Probab. Math. Stat."},{"key":"10.1016\/j.sysconle.2026.106514_b40","series-title":"Stochastic Partial Differential Equations and their Applications (Charlotte, NC, 1991)","first-page":"88","author":"M\u00e9l\u00e9ard","year":"1992"},{"key":"10.1016\/j.sysconle.2026.106514_b41","series-title":"Tightness of probabilities on C([0,1];S\u2032) and D([0,1];S\u2032)","first-page":"989","author":"Mitoma","year":"1983"},{"key":"10.1016\/j.sysconle.2026.106514_b42","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1080\/17442508008833147","article-title":"The variational principle and stochastic optimal control","volume":"3","author":"Elliott","year":"1980","journal-title":"Stochastics"},{"key":"10.1016\/j.sysconle.2026.106514_b43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/1467-9965.00022","article-title":"Backward stochastic differential equations in finance","volume":"7","author":"Karoui","year":"1997","journal-title":"Math. Finance"},{"key":"10.1016\/j.sysconle.2026.106514_b44","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1016\/j.jde.2021.09.012","article-title":"Backward stochastic differential equations with Markov chains and associated PDEs","volume":"302","author":"Ma","year":"2021","journal-title":"J. Differential Equations"},{"key":"10.1016\/j.sysconle.2026.106514_b45","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1016\/S0764-4442(97)87886-X","article-title":"A stability theorem of backward stochastic differential equations and its application","volume":"324","author":"Hu","year":"1997","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."}],"container-title":["Systems & Control Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S016769112600174X?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S016769112600174X?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T22:52:24Z","timestamp":1783205544000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S016769112600174X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9]]},"references-count":45,"alternative-id":["S016769112600174X"],"URL":"https:\/\/doi.org\/10.1016\/j.sysconle.2026.106514","relation":{},"ISSN":["0167-6911"],"issn-type":[{"value":"0167-6911","type":"print"}],"subject":[],"published":{"date-parts":[[2026,9]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"The stochastic maximum principle for mean-field relaxed control problem with regime-switching","name":"articletitle","label":"Article Title"},{"value":"Systems & Control Letters","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.sysconle.2026.106514","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"106514"}}