{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:09:48Z","timestamp":1760148588946,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,5,24]],"date-time":"2023-05-24T00:00:00Z","timestamp":1684886400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key R&amp;D Program of China","award":["2018Y-FA0703900","11871309","11371226","ZR2020QA026","ZR2020MA032","ZR2022MA029"],"award-info":[{"award-number":["2018Y-FA0703900","11871309","11371226","ZR2020QA026","ZR2020MA032","ZR2022MA029"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2018Y-FA0703900","11871309","11371226","ZR2020QA026","ZR2020MA032","ZR2022MA029"],"award-info":[{"award-number":["2018Y-FA0703900","11871309","11371226","ZR2020QA026","ZR2020MA032","ZR2022MA029"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["2018Y-FA0703900","11871309","11371226","ZR2020QA026","ZR2020MA032","ZR2022MA029"],"award-info":[{"award-number":["2018Y-FA0703900","11871309","11371226","ZR2020QA026","ZR2020MA032","ZR2022MA029"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, a general class of time-symmetric mean-field stochastic systems, namely the so-called mean-field forward-backward doubly stochastic differential equations (mean-field FBDSDEs, in short) are studied, where coefficients depend not only on the solution processes but also on their law. We first verify the existence and uniqueness of solutions for the forward equation of general mean-field FBDSDEs under Lipschitz conditions, and we obtain the associated comparison theorem; similarly, we also verify those results about the backward equation. As the above two comparison theorems\u2019 application, we prove the existence of the maximal solution for general mean-field FBDSDEs under some much weaker monotone continuity conditions. Furthermore, under appropriate assumptions we prove the uniqueness of the solution for the equations. Finally, we also obtain a comparison theorem for coupled general mean-field FBDSDEs.<\/jats:p>","DOI":"10.3390\/sym15061143","type":"journal-article","created":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T01:36:28Z","timestamp":1684978588000},"page":"1143","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["General Time-Symmetric Mean-Field Forward-Backward Doubly Stochastic Differential Equations"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6077-2550","authenticated-orcid":false,"given":"Nana","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Mathematics, Shandong University, Jinan 250100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6288-9851","authenticated-orcid":false,"given":"Jinghan","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute for Financial Studies, Shandong University, Jinan 250100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8784-8239","authenticated-orcid":false,"given":"Yufeng","family":"Shi","sequence":"additional","affiliation":[{"name":"Institute for Financial Studies, Shandong University, Jinan 250100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1262-9688","authenticated-orcid":false,"given":"Qingfeng","family":"Zhu","sequence":"additional","affiliation":[{"name":"Shandong Key Laboratory of Blockchain Finance, Shandong University of Finance and Economics, Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/0167-6911(90)90082-6","article-title":"Adapted solution of a backward stochastic differential equation","volume":"14","author":"Pardoux","year":"1990","journal-title":"Syst. 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