{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,5]],"date-time":"2025-07-05T10:45:16Z","timestamp":1751712316320,"version":"3.37.3"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,7,24]],"date-time":"2024-07-24T00:00:00Z","timestamp":1721779200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,24]],"date-time":"2024-07-24T00:00:00Z","timestamp":1721779200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100019239","name":"Berlin Mathematics Research Center MATH+","doi-asserted-by":"publisher","award":["EF4-6"],"award-info":[{"award-number":["EF4-6"]}],"id":[{"id":"10.13039\/501100019239","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100022023","name":"Brandenburgische Technische Universit\u00e4t Cottbus - Senftenberg","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100022023","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we study a system of decoupled forward-backward stochastic differential equations driven by a <jats:italic>G<\/jats:italic>-Brownian motion (<jats:italic>G<\/jats:italic>-FBSDEs) with non-degenerate diffusion. Our objective is to establish the existence of a relaxed optimal control for a non-smooth stochastic optimal control problem. The latter is given in terms of a decoupled <jats:italic>G<\/jats:italic>-FBSDE. The cost functional is the solution of the backward stochastic differential equation at the initial time. The key idea to establish existence of a relaxed optimal control is to replace the original control problem by a suitably regularised problem with mollified coefficients, prove the existence of a relaxed control, and then pass to the limit.\n<\/jats:p>","DOI":"10.1007\/s10957-024-02495-2","type":"journal-article","created":{"date-parts":[[2024,7,24]],"date-time":"2024-07-24T15:10:19Z","timestamp":1721833819000},"page":"1027-1059","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimal Relaxed Control for a Decoupled G-FBSDE"],"prefix":"10.1007","volume":"202","author":[{"given":"Hafida","family":"Bouanani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1420-2071","authenticated-orcid":false,"given":"Omar","family":"Kebiri","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carsten","family":"Hartmann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amel","family":"Redjil","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,7,24]]},"reference":[{"issue":"6","key":"2495_CR1","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1080\/17442508.2018.1427750","volume":"90","author":"K Bahlali","year":"2018","unstructured":"Bahlali, K., Kebiri, O., Mezerdi, B., Mtiraoui, A.: Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient. Stochastics 90(6), 861\u2013875 (2018)","journal-title":"Stochastics"},{"issue":"3","key":"2495_CR2","doi-asserted-by":"publisher","first-page":"589","DOI":"10.1007\/s10255-014-0405-9","volume":"30","author":"X Bai","year":"2014","unstructured":"Bai, X., Lin, Y.: On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients. Acta Mathematicae Applicatae Sinica English Ser. 30(3), 589\u2013610 (2014)","journal-title":"Acta Mathematicae Applicatae Sinica English Ser."},{"issue":"2","key":"2495_CR3","doi-asserted-by":"publisher","first-page":"384","DOI":"10.1016\/0022-247X(73)90066-8","volume":"44","author":"J-M Bismut","year":"1973","unstructured":"Bismut, J.-M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44(2), 384\u2013404 (1973)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"2495_CR4","doi-asserted-by":"publisher","first-page":"827","DOI":"10.1214\/105051606000000169","volume":"16","author":"L Denis","year":"2006","unstructured":"Denis, L., Martini, C., et al.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16(2), 827\u2013852 (2006)","journal-title":"Ann. Appl. Probab."},{"issue":"1","key":"2495_CR5","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), 1\u201371 (1997)","journal-title":"Math. Financ."},{"key":"2495_CR6","doi-asserted-by":"publisher","first-page":"692","DOI":"10.5560\/zna.2012-0092","volume":"67a","author":"F Faizullah","year":"2012","unstructured":"Faizullah, F.: Existence of solutions for stochastic differential equations under G-Brownian motion with discontinuous coefficients. Zeitschrift f\u00fcr Naturforschung A 67a, 692\u2013698, 05 (2012)","journal-title":"Zeitschrift f\u00fcr Naturforschung A"},{"issue":"1","key":"2495_CR7","doi-asserted-by":"publisher","first-page":"759","DOI":"10.1016\/j.spa.2013.09.010","volume":"124","author":"M Hu","year":"2014","unstructured":"Hu, M., Ji, S., Peng, S., Song, Y.: Backward stochastic differential equations driven by G-Brownian motion. Stoch. Proc. Appl. 124(1), 759\u2013784 (2014)","journal-title":"Stoch. Proc. Appl."},{"key":"2495_CR8","doi-asserted-by":"crossref","unstructured":"Krylov, N.V.: Nonlinear elliptic and parabolic equations of the second order, vol. 7. Springer, Nikolaj Vladimirovi\u010d Krylov (1987)","DOI":"10.1007\/978-94-010-9557-0"},{"issue":"1","key":"2495_CR9","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/j.spa.2016.06.002","volume":"127","author":"H Mingshang","year":"2017","unstructured":"Mingshang, H., Ji, S.: Dynamic programming principle for stochastic recursive optimal control problem driven by a G-Brownian motion. Stoch. Process. Appl. 127(1), 107\u2013134 (2017)","journal-title":"Stoch. Process. Appl."},{"issue":"2","key":"2495_CR10","doi-asserted-by":"publisher","first-page":"1170","DOI":"10.1016\/j.spa.2013.10.009","volume":"124","author":"H Mingshang","year":"2014","unstructured":"Mingshang, H., Ji, S., Peng, S., Song, Y.: Comparison theorem, Feynman\u2013Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion. Stoch. Proc. Appl. 124(2), 1170\u20131195 (2014)","journal-title":"Stoch. Proc. Appl."},{"key":"2495_CR11","first-page":"84","volume":"355","author":"A Mtiraoui","year":"2017","unstructured":"Mtiraoui, A., Bahlali, K., Kebiri, O.: Existence of an optimal control for a system driven by a degenerate coupled forward-backward stochastic differential equations. C. R. Math. 355, 84\u201389 (2017)","journal-title":"C. R. Math."},{"issue":"4","key":"2495_CR12","doi-asserted-by":"publisher","first-page":"966","DOI":"10.1137\/0328054","volume":"28","author":"S Peng","year":"1990","unstructured":"Peng, S.: A general stochastic maximum principle for optimal control problems. SIAM J. Control. Optim. 28(4), 966\u2013979 (1990)","journal-title":"SIAM J. Control. Optim."},{"issue":"1\u20132","key":"2495_CR13","first-page":"61","volume":"37","author":"S Peng","year":"1991","unstructured":"Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 37(1\u20132), 61\u201374 (1991)","journal-title":"Stoch. Stoch. Rep."},{"issue":"2","key":"2495_CR14","first-page":"119","volume":"38","author":"S Peng","year":"1992","unstructured":"Peng, S.: A generalized dynamic programming principle and Hamilton\u2013Jacobi\u2013Bellman equation. Stochastics 38(2), 119\u2013134 (1992)","journal-title":"Stochastics"},{"key":"2495_CR15","unstructured":"Peng, S.: G-Brownian motion and dynamic risk measure under volatility uncertainty. arXiv e-prints arXiv:0711.2834 (2007)"},{"key":"2495_CR16","doi-asserted-by":"crossref","unstructured":"Peng, S.: Nonlinear Expectations and Stochastic Calculus Under Uncertainty: With Robust CLT and G-Brownian Motion, volume\u00a095. Springer (2019)","DOI":"10.1007\/978-3-662-59903-7"},{"issue":"1","key":"2495_CR17","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/0167-6911(90)90082-6","volume":"14","author":"E Pardoux","year":"1990","unstructured":"Pardoux, E., Peng, S.G.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55\u201361 (1990)","journal-title":"Syst. Control Lett."},{"issue":"1","key":"2495_CR18","first-page":"201","volume":"15","author":"A Redjil","year":"2018","unstructured":"Redjil, A., Choutri, S.E.: On relaxed stochastic optimal control for stochastic differential equations driven by G-Brownian motion. ALEA Lat. Am. J. Probab. 15(1), 201\u2013212 (2018)","journal-title":"ALEA Lat. Am. J. Probab."},{"key":"2495_CR19","volume-title":"Linear and Quasilinear Equations of Parabolic Type","author":"VA Solonnikov","year":"1968","unstructured":"Solonnikov, V.A., Ladyzhenskaya, O.A., Ural\u2019tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS, Rhode Island (1968)"},{"key":"2495_CR20","first-page":"39","volume":"349","author":"B Wang","year":"2019","unstructured":"Wang, B., Yuan, M.: Forward-backward stochastic differential equations driven by G-Brownian motion. Appl. Math. Comput. 349, 39\u201347 (2019)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"2495_CR21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-016-1057-2","volume":"2017","author":"B Wang","year":"2017","unstructured":"Wang, B., Yuan, M.: Existence of solution for stochastic differential equations driven by G-L\u00e9vy process with discontinuous coefficients. Adv. Differ. Equ. 2017(1), 1\u201313 (2017)","journal-title":"Adv. Differ. Equ."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02495-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-024-02495-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02495-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,4]],"date-time":"2024-09-04T14:05:36Z","timestamp":1725458736000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-024-02495-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,24]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["2495"],"URL":"https:\/\/doi.org\/10.1007\/s10957-024-02495-2","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2024,7,24]]},"assertion":[{"value":"29 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 July 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}