{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T04:36:30Z","timestamp":1772512590847,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,8,31]],"date-time":"2024-08-31T00:00:00Z","timestamp":1725062400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,8,31]],"date-time":"2024-08-31T00:00:00Z","timestamp":1725062400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"National Natural Science Foundation of China, China Postdoctoral Science Foundation, China Postdoctoral Innovation Talent Support Program, Shandong Postdoctoral Science Foundation","award":["No. 12301574, No. 2023M732085, No. BX20230206, No. SDCX-ZG-202301005"],"award-info":[{"award-number":["No. 12301574, No. 2023M732085, No. BX20230206, No. SDCX-ZG-202301005"]}]},{"name":"National Natural Science Foundation of China, Taishan Scholars Young Program of Shandong, Young Scholars Program of Shandong University","award":["No.TSQN202211032"],"award-info":[{"award-number":["No.TSQN202211032"]}]},{"name":"National Key R &D Program of China, National Natural Science Foundation of China, Natural Science Foundation of Shandong Province","award":["No. 2023YFA1009203, No. 12371447, No. ZR2022JQ01"],"award-info":[{"award-number":["No. 2023YFA1009203, No. 12371447, No. ZR2022JQ01"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1007\/s10957-024-02508-0","type":"journal-article","created":{"date-parts":[[2024,8,31]],"date-time":"2024-08-31T03:44:15Z","timestamp":1725075855000},"page":"2487-2508","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Incomplete Information Mean-Field Games and Related Riccati Equations"],"prefix":"10.1007","volume":"203","author":[{"given":"Min","family":"Li","sequence":"first","affiliation":[]},{"given":"Tianyang","family":"Nie","sequence":"additional","affiliation":[]},{"given":"Shujun","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6937-5618","authenticated-orcid":false,"given":"Ke","family":"Yan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,31]]},"reference":[{"key":"2508_CR1","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1016\/0016-0032(83)90059-5","volume":"315","author":"A Bensoussan","year":"1983","unstructured":"Bensoussan, A.: Stochastic maximum principle for distributed parameter systems. J. Franklin I. 315, 387\u2013406 (1983)","journal-title":"J. Franklin I."},{"key":"2508_CR2","doi-asserted-by":"publisher","first-page":"23","DOI":"10.3934\/mcrf.2020025","volume":"11","author":"A Bensoussan","year":"2021","unstructured":"Bensoussan, A., Feng, X., Huang, J.: Linear-Quadratic-Gaussian mean-field-game with partial observation and common noise. Math. Control Relat. Fields. 11, 23\u201346 (2021)","journal-title":"Math. Control Relat. Fields."},{"key":"2508_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-8508-7","volume-title":"Mean Field Games and Mean Field Type Control Theory","author":"A Bensoussan","year":"2013","unstructured":"Bensoussan, A., Frehse, J., Yam, P.: Mean Field Games and Mean Field Type Control Theory. Springer, Springer, New York (2013)"},{"issue":"4","key":"2508_CR4","doi-asserted-by":"publisher","first-page":"2840","DOI":"10.1287\/moor.2021.1230","volume":"47","author":"E Bayraktar","year":"2022","unstructured":"Bayraktar, E., Cecchin, A., Cohen, A., Delarue, F.: Finite state mean field games with wright-fisher common noise as limits of N-player weighted games. Math. Oper. Res. 47(4), 2840\u20132890 (2022)","journal-title":"Math. Oper. Res."},{"key":"2508_CR5","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1016\/j.spa.2021.01.003","volume":"134","author":"R Buckdahn","year":"2021","unstructured":"Buckdahn, R., Chen, Y., Li, J.: Partial derivative with respect to the measure and its applications to general controlled mean-field systems. Stochatic Process. Appl. 134, 265\u2013307 (2021)","journal-title":"Stochatic Process. Appl."},{"key":"2508_CR6","doi-asserted-by":"publisher","first-page":"3201","DOI":"10.1214\/17-AAP1280","volume":"27","author":"R Buckdahn","year":"2017","unstructured":"Buckdahn, R., Li, J., Ma, J.: A mean-field stochastic control problem with partial observations. Ann. Appl. Probab. 27, 3201\u20133245 (2017)","journal-title":"Ann. Appl. Probab."},{"key":"2508_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-56436-4","volume-title":"Probabilistic Theory of Mean-field Games with Applications","author":"R Carmona","year":"2018","unstructured":"Carmona, R., Delarue, F.: Probabilistic Theory of Mean-field Games with Applications. Springer, New York (2018)"},{"key":"2508_CR8","doi-asserted-by":"publisher","first-page":"3740","DOI":"10.1214\/15-AOP1060","volume":"44","author":"R Carmona","year":"2016","unstructured":"Carmona, R., Delarue, F., Lacker, D.: Mean field games with common noise. Ann. Probab. 44, 3740\u20133803 (2016)","journal-title":"Ann. Probab."},{"key":"2508_CR9","doi-asserted-by":"publisher","first-page":"911","DOI":"10.4310\/CMS.2015.v13.n4.a4","volume":"13","author":"R Carmona","year":"2015","unstructured":"Carmona, R., Fouque, J., Sun, L.: Mean field games and systemic risk. Commun. Math. Sci. 13, 911\u2013933 (2015)","journal-title":"Commun. Math. Sci."},{"key":"2508_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1111\/1467-9965.00022","volume":"7","author":"N El Karoui","year":"1997","unstructured":"El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Financ. 7, 1\u201371 (1997)","journal-title":"Math. Financ."},{"key":"2508_CR11","unstructured":"Huang, M., Caines, P., Malham\u00e9, R.: Distributed multi-agent decision-making with partial observations: asymtotic Nash equilibria. In: Proc. the 17th internat. Symposium on math. Theory on networks and systems, Kyoto, Japan, (2006)"},{"key":"2508_CR12","doi-asserted-by":"publisher","first-page":"2835","DOI":"10.1137\/17M1151420","volume":"56","author":"Y Hu","year":"2018","unstructured":"Hu, Y., Huang, J., Nie, T.: Linear-Quadratic-Gaussian mixed mean-field games with heterogeneous input constraints. SIAM J. Control. Optim. 56, 2835\u20132877 (2018)","journal-title":"SIAM J. Control. Optim."},{"key":"2508_CR13","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1007\/s11424-007-9015-4","volume":"20","author":"M Huang","year":"2007","unstructured":"Huang, M., Caines, P., Malham\u00e9, R.: An invariance principle in large population stochastic dynamic games. J. Sys. Sci. Complex. 20, 162\u2013172 (2007)","journal-title":"J. Sys. Sci. Complex."},{"key":"2508_CR14","doi-asserted-by":"publisher","first-page":"221","DOI":"10.4310\/CIS.2006.v6.n3.a5","volume":"6","author":"M Huang","year":"2006","unstructured":"Huang, M., Malham\u00e9, R., Caines, P.: Large population stochastic dynamic games: closed-loop McKean\u2013Vlasov systems and the Nash certainty equivalence principle. Comm. Inform. Systems 6, 221\u2013251 (2006)","journal-title":"Comm. Inform. Systems"},{"key":"2508_CR15","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/s10957-015-0740-x","volume":"168","author":"J Huang","year":"2016","unstructured":"Huang, J., Wang, S.: Dynamic optimization of large-population systems with partial information. J. Optim. Theory Appl. 168, 231\u2013245 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"2508_CR16","doi-asserted-by":"publisher","first-page":"3784","DOI":"10.1109\/TAC.2016.2519501","volume":"61","author":"J Huang","year":"2016","unstructured":"Huang, J., Wang, S., Wu, Z.: Backward mean-field Linear-Quadratic-Gaussian (LQG) games: full and partial information. IEEE Trans. Automat. Control 61, 3784\u20133796 (2016)","journal-title":"IEEE Trans. Automat. Control"},{"key":"2508_CR17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s41546-016-0001-4","volume":"1","author":"J Huang","year":"2016","unstructured":"Huang, J., Wang, S., Wu, Z.: Backward-forward linear-quadratic mean-field games with major and minor agents. Probab. Uncertaint. Qua. 1, 1\u201327 (2016)","journal-title":"Probab. Uncertaint. Qua."},{"key":"2508_CR18","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1007\/s10915-022-01979-5","volume":"93","author":"S Ji","year":"2022","unstructured":"Ji, S., Peng, S., Peng, Y., Zhang, X.: Solving stochastic optimal control problem via stochastic maximum principle with deep learning method. J. Sci. Comput. 93, 28 (2022)","journal-title":"J. Sci. Comput."},{"key":"2508_CR19","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s11537-007-0657-8","volume":"2","author":"J Lasry","year":"2007","unstructured":"Lasry, J., Lions, P.: Mean field games. Japan. J. Math. 2, 229\u2013260 (2007)","journal-title":"Japan. J. Math."},{"key":"2508_CR20","doi-asserted-by":"publisher","first-page":"2463","DOI":"10.1080\/00207179.2018.1441555","volume":"92","author":"R Li","year":"2019","unstructured":"Li, R., Fu, F.: The maximum principle for partially observed optimal control problems of mean-field FBSDEs. Int. J. Control 92, 2463\u20132472 (2019)","journal-title":"Int. J. Control"},{"issue":"6","key":"2508_CR21","doi-asserted-by":"publisher","first-page":"4105","DOI":"10.1090\/tran\/8868","volume":"376","author":"M Li","year":"2023","unstructured":"Li, M., Mou, C., Wu, Z., Zhou, C.: Linear-quadratic mean-filed games of controls with Non-Monotone Data. Trans. Amer. Math. Soc. 376(6), 4105\u20134143 (2023)","journal-title":"Trans. Amer. Math. Soc."},{"issue":"4","key":"2508_CR22","doi-asserted-by":"publisher","first-page":"2114","DOI":"10.1137\/21M1414152","volume":"61","author":"M Li","year":"2023","unstructured":"Li, M., Nie, T., Wu, Z.: Linear-quadratic large-population problem with partial information: Hamiltonian approach and Riccati approach. SIAM J. Control. Optim. 61(4), 2114\u20132139 (2023)","journal-title":"SIAM J. Control. Optim."},{"key":"2508_CR23","unstructured":"Li, M., Nie, T., Wang, S., Yan, K.: Incomplete information linear-quadratic mean-field games and related Riccati equations. arXiv:2307.01005, (2023)"},{"key":"2508_CR24","doi-asserted-by":"publisher","first-page":"1062","DOI":"10.1109\/TAC.2016.2579264","volume":"62","author":"J Moon","year":"2017","unstructured":"Moon, J., Ba\u015far, T.: Linear quadratic risk-sensitive and robust mean field games. IEEE Trans. Automat. Control 62, 1062\u20131077 (2017)","journal-title":"IEEE Trans. Automat. Control"},{"key":"2508_CR25","first-page":"143","volume":"20","author":"D Majerek","year":"2005","unstructured":"Majerek, D., Nowak, W., Zi\u0229ba, W.: Conditional strong law of large number. Int. J. Pure Appl. Math. 20, 143\u2013156 (2005)","journal-title":"Int. J. Pure Appl. Math."},{"key":"2508_CR26","doi-asserted-by":"publisher","first-page":"2064","DOI":"10.1137\/17M1140133","volume":"57","author":"N \u015een","year":"2019","unstructured":"\u015een, N., Caines, P.: Mean field games with partial observation. SIAM J. Control. Optim. 57, 2064\u20132091 (2019)","journal-title":"SIAM J. Control. Optim."},{"key":"2508_CR27","doi-asserted-by":"publisher","first-page":"243","DOI":"10.4208\/nmtma.OA-2023-0048","volume":"17","author":"Y Sun","year":"2024","unstructured":"Sun, Y., Yang, J., Zhao, W.: An accurate numerical scheme for mean-field forward and backward SDEs with jumps. Numer. Math. Theory Me. 17, 243\u2013274 (2024)","journal-title":"Numer. Math. Theory Me."},{"key":"2508_CR28","doi-asserted-by":"crossref","unstructured":"Siwe, A.T., Tembine, H.: Network security as public good: A mean-field-type game theory approach. In: Proc. 13th Int. Multiconf. Syst. Singles Devices (SSD), pp. 601-606 (2016)","DOI":"10.1109\/SSD.2016.7473659"},{"key":"2508_CR29","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s13235-016-0200-8","volume":"8","author":"RF Tchuendom","year":"2018","unstructured":"Tchuendom, R.F.: Uniqueness for linear-quadratic mean field games with common noise. Dyn. Games. Appl. 8, 199\u2013210 (2018)","journal-title":"Dyn. Games. Appl."},{"key":"2508_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-79039-8","volume-title":"An Introduction to Optimal Control of FBSDE with Incomplete Information","author":"G Wang","year":"2018","unstructured":"Wang, G., Wu, Z., Xiong, J.: An Introduction to Optimal Control of FBSDE with Incomplete Information. Springer, Cham (2018)"},{"key":"2508_CR31","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1137\/15M104178X","volume":"55","author":"B Wang","year":"2017","unstructured":"Wang, B., Zhang, J.: Social optima in mean field Linear-Quadratic-Gaussian models with Markov jump parameters. SIAM J. Control. Optim. 55, 429\u2013456 (2017)","journal-title":"SIAM J. Control. Optim."},{"key":"2508_CR32","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1016\/j.automatica.2017.07.018","volume":"86","author":"G Wang","year":"2017","unstructured":"Wang, G., Xiao, H., Xing, G.: An optimal control problem for mean-field forward-backward stochastic differential equation with noisy observation. Automatica 86, 104\u2013109 (2017)","journal-title":"Automatica"},{"key":"2508_CR33","doi-asserted-by":"publisher","first-page":"522","DOI":"10.1109\/TAC.2013.2273265","volume":"59","author":"G Wang","year":"2013","unstructured":"Wang, G., Zhang, C., Zhang, W.: Stochastic maximum principle for mean-field type optimal control under partial information. IEEE Trans. Automat. Control 59, 522\u2013528 (2013)","journal-title":"IEEE Trans. Automat. Control"},{"key":"2508_CR34","doi-asserted-by":"publisher","first-page":"2809","DOI":"10.1137\/120892477","volume":"51","author":"J Yong","year":"2013","unstructured":"Yong, J.: Linear-quadratic optimal control problems for mean-field stochastic differential equations. SIAM J. Control. Optim. 51, 2809\u20132838 (2013)","journal-title":"SIAM J. Control. Optim."},{"key":"2508_CR35","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1466-3","volume-title":"Stochastic Controls: Hamiltonian Systems and HJB Equations","author":"J Yong","year":"1999","unstructured":"Yong, J., Zhou, X.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)"},{"issue":"11","key":"2508_CR36","doi-asserted-by":"publisher","first-page":"11805","DOI":"10.1109\/TCYB.2021.3070352","volume":"52","author":"W Zhang","year":"2021","unstructured":"Zhang, W., Peng, C.: Indefinite mean-field stochastic cooperative linear-quadratic dynamic difference game with its application to the network security model. IEEE Trans. Cybern. 52(11), 11805\u201311818 (2021)","journal-title":"IEEE Trans. Cybern."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02508-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-024-02508-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02508-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T20:06:25Z","timestamp":1733774785000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-024-02508-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,31]]},"references-count":36,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["2508"],"URL":"https:\/\/doi.org\/10.1007\/s10957-024-02508-0","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,31]]},"assertion":[{"value":"13 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 August 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}