{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T12:36:53Z","timestamp":1740141413585,"version":"3.37.3"},"reference-count":31,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T00:00:00Z","timestamp":1651190400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971185"],"award-info":[{"award-number":["11971185"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,6,24]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This paper is concerned with the first-order and second-order necessary optimality conditions for discrete-time stochastic systems with delay under weak assumptions. Based on a characteristic method and a discrete-time backward stochastic equation, we establish the linearized discrete-time stochastic maximum principle and Euler-type necessary optimality condition. Moreover, by a new discrete-time backward stochastic matrix equation, we also obtain the necessary optimality condition of quasi-singular control and the pointwise second-order optimality condition.<\/jats:p>","DOI":"10.1093\/imamci\/dnac010","type":"journal-article","created":{"date-parts":[[2022,5,4]],"date-time":"2022-05-04T19:11:01Z","timestamp":1651691461000},"page":"731-750","source":"Crossref","is-referenced-by-count":1,"title":["First-order and second-order necessary optimality conditions for discrete-time stochastic systems with delay"],"prefix":"10.1093","volume":"39","author":[{"given":"Teng","family":"Song","sequence":"first","affiliation":[{"name":"School of Science , , 430070 Wuhan , P R China"},{"name":"Wuhan University of Technology , , 430070 Wuhan , P R China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics , Hubei Key Laboratory of Engineering Modeling and Scientific Computing, , 430074 Wuhan , P R China"},{"name":"Huazhong University of Science and Technology , Hubei Key Laboratory of Engineering Modeling and Scientific Computing, , 430074 Wuhan , P R China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2022,4,29]]},"reference":[{"key":"2022062413513029600_ref1","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1016\/0016-0032(83)90059-5","article-title":"Stochastic maximum principle for distributed parameter systems","volume":"315","author":"Bensoussan","year":"1983","journal-title":"J. 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