{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T03:16:46Z","timestamp":1766546206981,"version":"3.37.3"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,8,1]],"date-time":"2023-08-01T00:00:00Z","timestamp":1690848000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,8,1]],"date-time":"2023-08-01T00:00:00Z","timestamp":1690848000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000015","name":"The U.S. Department of Energy","doi-asserted-by":"crossref","award":["DE-0022257"],"award-info":[{"award-number":["DE-0022257"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2023,8]]},"DOI":"10.1007\/s10444-023-10053-z","type":"journal-article","created":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T08:02:29Z","timestamp":1691049749000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Splitting scheme for backward doubly stochastic differential equations"],"prefix":"10.1007","volume":"49","author":[{"given":"Feng","family":"Bao","sequence":"first","affiliation":[]},{"given":"Yanzhao","family":"Cao","sequence":"additional","affiliation":[]},{"given":"He","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,8,3]]},"reference":[{"key":"10053_CR1","first-page":"592","volume":"4","author":"A Bachouch","year":"2016","unstructured":"Bachouch, A., Ben Lasmar, M.A., Matoussi, A., Mnif, M.: Euler time discretization of backward doubly SDEs and application to semilinear SPDEs. Stoch. Partial Differ. Equ.: Anal. Comput. 4, 592\u2013634 (2016)","journal-title":"Stoch. Partial Differ. Equ.: Anal. Comput."},{"key":"10053_CR2","doi-asserted-by":"publisher","first-page":"635","DOI":"10.4310\/CMS.2020.v18.n3.a3","volume":"18","author":"F Bao","year":"2020","unstructured":"Bao, F., Cao, Y., Han, X.: Forward backward doubly stochastic differential equations and the optimal filtering of diffusion processes. Commun. Math. Sci. 18, 635\u2013661 (2020)","journal-title":"Commun. Math. Sci."},{"key":"10053_CR3","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1137\/14095546X","volume":"4","author":"F Bao","year":"2016","unstructured":"Bao, F., Cao, Y., Meir, A., Zhao, W.: A first order scheme for backward doubly stochastic differential equations. SIAM\/ASA Journal on Uncertainty Quantification 4, 413\u2013445 (2016)","journal-title":"SIAM\/ASA Journal on Uncertainty Quantification"},{"key":"10053_CR4","doi-asserted-by":"publisher","first-page":"A2664","DOI":"10.1137\/16M1100277","volume":"39","author":"F Bao","year":"2017","unstructured":"Bao, F., Maroulas, V.: Adaptive meshfree backward SDE filter. SIAM J. Sci. Comput. 39, A2664\u2013A2683 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"10053_CR5","doi-asserted-by":"crossref","unstructured":"Bensoussan, A., Glowinski, R.: Approximation of Zakai equation by the splitting up method. In: Stochastic Systems and Optimization, pp.\u00a0255\u2013265. Springer (1989)","DOI":"10.1007\/BFb0002686"},{"key":"10053_CR6","doi-asserted-by":"publisher","first-page":"1420","DOI":"10.1137\/0328074","volume":"28","author":"A Bensoussan","year":"1990","unstructured":"Bensoussan, A., Glowinski, R., R\u0103\u015fcanu, A.: Approximation of the Zakai equation by the splitting up method. SIAM J. Control. Optim. 28, 1420\u20131431 (1990)","journal-title":"SIAM J. Control. Optim."},{"key":"10053_CR7","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/BF01184157","volume":"25","author":"A Bensoussan","year":"1992","unstructured":"Bensoussan, A., Glowinski, R., R\u0103\u015fcanu, A.: Approximation of some stochastic differential equations by the splitting up method. Appl. Math. Optim. 25, 81\u2013106 (1992)","journal-title":"Appl. Math. Optim."},{"key":"10053_CR8","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s40304-017-0117-6","volume":"5","author":"E Weinan","year":"2017","unstructured":"Weinan, E., Han, J., Jentzen, A.: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat. 5, 349\u2013380 (2017)","journal-title":"Commun. Math. Stat."},{"key":"10053_CR9","doi-asserted-by":"publisher","first-page":"C652","DOI":"10.1137\/16M106371X","volume":"38","author":"E Gobet","year":"2016","unstructured":"Gobet, E., L\u00f3pez-Salas, J.G., Turkedjiev, P., V\u00e1zquez, C.: Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs. SIAM J. Sci. Comput. 38, C652\u2013C677 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"10053_CR10","doi-asserted-by":"publisher","first-page":"1171","DOI":"10.1016\/j.spa.2016.07.011","volume":"127","author":"E Gobet","year":"2017","unstructured":"Gobet, E., Turkedjiev, P.: Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations. Stochastic Processes and their Applications 127, 1171\u20131203 (2017)","journal-title":"Stochastic Processes and their Applications"},{"key":"10053_CR11","doi-asserted-by":"publisher","first-page":"564","DOI":"10.1214\/aop\/1048516528","volume":"31","author":"I Gy\u00f6ngy","year":"2003","unstructured":"Gy\u00f6ngy, I., Krylov, N.: On the splitting-up method and stochastic partial differential equations. Ann. Probab. 31, 564\u2013591 (2003)","journal-title":"Ann. Probab."},{"key":"10053_CR12","unstructured":"Kloeden, P., Platen, E.: Numerical solution of stochastic differential equations, vol.\u00a023. Springer Science & Business Media (2013)"},{"key":"10053_CR13","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1515\/mcma-2013-0001","volume":"19","author":"C Labart","year":"2013","unstructured":"Labart, C., Lelong, J.: A parallel algorithm for solving BSDEs. Monte Carlo Methods and Applications 19, 11\u201339 (2013)","journal-title":"Monte Carlo Methods and Applications"},{"key":"10053_CR14","doi-asserted-by":"crossref","unstructured":"LeGland, F.: Splitting-up approximation for SPDE\u2019s and SDE\u2019s with application to nonlinear filtering. In: Stochastic partial differential equations and their applications, pp.\u00a0177\u2013187. Springer (1992)","DOI":"10.1007\/BFb0007332"},{"key":"10053_CR15","first-page":"302","volume":"12","author":"J Ma","year":"2002","unstructured":"Ma, J., Protter, P., San Martin, J., Torres, S., et al.: Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12, 302\u2013316 (2002)","journal-title":"Ann. Appl. Probab."},{"key":"10053_CR16","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1007\/BF01192258","volume":"98","author":"J Ma","year":"1994","unstructured":"Ma, J., Protter, P., Yong, J.: Solving forward-backward stochastic differential equations explicitly\u2013a four step scheme. Probab. Theory Relat. Fields 98, 339\u2013359 (1994)","journal-title":"Probab. Theory Relat. Fields"},{"key":"10053_CR17","doi-asserted-by":"crossref","unstructured":"Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and their Applications, pp.\u00a0200\u2013217. Springer (1992)","DOI":"10.1007\/BFb0007334"},{"key":"10053_CR18","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1007\/BF01192514","volume":"98","author":"E Pardoux","year":"1994","unstructured":"Pardoux, E., Peng, S.: Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Relat. Fields 98, 209\u2013227 (1994)","journal-title":"Probab. Theory Relat. Fields"},{"key":"10053_CR19","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/BF00390274","volume":"76","author":"E Pardoux","year":"1987","unstructured":"Pardoux, E., Protter, P.: A two-sided stochastic integral and its calculus. Probab. Theory Relat. Fields 76, 15\u201349 (1987)","journal-title":"Probab. Theory Relat. Fields"},{"key":"10053_CR20","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1007\/BF00536382","volume":"11","author":"M Zakai","year":"1969","unstructured":"Zakai, M.: On the optimal filtering of diffusion processes. Zeitschrift f\u00fcr Wahrscheinlichkeitstheorie und verwandte Gebiete 11, 230\u2013243 (1969)","journal-title":"Zeitschrift f\u00fcr Wahrscheinlichkeitstheorie und verwandte Gebiete"},{"key":"10053_CR21","doi-asserted-by":"publisher","first-page":"221","DOI":"10.4208\/jcm.1212-m4014","volume":"31","author":"G Zhang","year":"2013","unstructured":"Zhang, G., Gunzburger, M., Zhao, W.: A sparse-grid method for multi-dimensional backward stochastic differential equations. J. Comput. Math. 31, 221\u2013248 (2013)","journal-title":"J. Comput. Math."},{"key":"10053_CR22","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1214\/aoap\/1075828058","volume":"14","author":"J Zhang","year":"2004","unstructured":"Zhang, J.: A numerical scheme for BSDEs. Ann. Appl. Probab. 14, 459\u2013488 (2004)","journal-title":"Ann. Appl. Probab."},{"key":"10053_CR23","doi-asserted-by":"crossref","unstructured":"Zhang, J.: Backward stochastic differential equations. In: Backward Stochastic Differential Equations, pp.\u00a079\u201399. Springer (2017)","DOI":"10.1007\/978-1-4939-7256-2_4"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-023-10053-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-023-10053-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-023-10053-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,22]],"date-time":"2023-08-22T07:13:19Z","timestamp":1692688399000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-023-10053-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8]]},"references-count":23,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["10053"],"URL":"https:\/\/doi.org\/10.1007\/s10444-023-10053-z","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"type":"print","value":"1019-7168"},{"type":"electronic","value":"1572-9044"}],"subject":[],"published":{"date-parts":[[2023,8]]},"assertion":[{"value":"24 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 June 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 August 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"65"}}