{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:42:57Z","timestamp":1760233377483,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T00:00:00Z","timestamp":1610409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871309, 11671229, 11371226, 11301298"],"award-info":[{"award-number":["11871309, 11671229, 11371226, 11301298"]}],"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":["ZR2019MA013"],"award-info":[{"award-number":["ZR2019MA013"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Key R&amp;D Program of China","award":["2018YFA0703900"],"award-info":[{"award-number":["2018YFA0703900"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward\u2013backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.<\/jats:p>","DOI":"10.3390\/sym13010118","type":"journal-article","created":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T20:11:31Z","timestamp":1610482291000},"page":"118","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Type of Time-Symmetric Stochastic System and Related Games"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1262-9688","authenticated-orcid":false,"given":"Qingfeng","family":"Zhu","sequence":"first","affiliation":[{"name":"School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China"},{"name":"Shandong Key Laboratory of Blockchain Finance, Shandong University of Finance and Economics, Jinan 250014, China"},{"name":"Institute for Financial Studies and School of Mathematics, 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 and School of Mathematics, Shandong University, Jinan 250100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiaqiang","family":"Wen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hui","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China"},{"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":[[2021,1,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/BF01192514","article-title":"Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDEs","volume":"98","author":"Pardoux","year":"1994","journal-title":"Probab. 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