{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T03:39:40Z","timestamp":1775014780117,"version":"3.50.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T00:00:00Z","timestamp":1619654400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T00:00:00Z","timestamp":1619654400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100012325","name":"Bergische Universit\u00e4t Wuppertal","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100012325","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we propose a new multi-step scheme by adopting the high-order multi-step method in Zhao et al. (SIAM J. Sci. Comput., 36(4): A1731-A1751, 2014) with the combination technique. Two reference ordinary differential equations containing the conditional expectations and their derivatives are derived from the backward component. These derivatives are approximated by using the finite difference methods with multi-step combinations. The resulting scheme is a semi-discretization in the temporal direction involving conditional expectations, which are solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for higher convergence rate up to ninth order, and are more efficient. Finally, we provide a numerical illustration of the convergence of the proposed method.<\/jats:p>","DOI":"10.1007\/s10915-021-01505-z","type":"journal-article","created":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T16:02:31Z","timestamp":1619712151000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["High-order Combined Multi-step Scheme for Solving Forward Backward Stochastic Differential Equations"],"prefix":"10.1007","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1540-621X","authenticated-orcid":false,"given":"Long","family":"Teng","sequence":"first","affiliation":[]},{"given":"Weidong","family":"Zhao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,29]]},"reference":[{"key":"1505_CR1","volume-title":"Handbook of Mathematical Functions, Dover Books on Mathematics","author":"M Abramowitz","year":"1972","unstructured":"Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions, Dover Books on Mathematics. Dover Publications, USA (1972)"},{"key":"1505_CR2","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/978-3-642-25746-9_8","volume":"12","author":"C Bender","year":"2012","unstructured":"Bender, C., Steiner, J.: Least-squares monte carlo for backward sdes. Numer. Methods Finance 12, 257\u2013289 (2012)","journal-title":"Numer. Methods Finance"},{"key":"1505_CR3","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1214\/07-AAP448","volume":"18","author":"C Bender","year":"2008","unstructured":"Bender, C., Zhang, J.: Time discretization and markovian iteration for coupled fbsdes. Ann. Appl. Probab. 18, 143\u2013177 (2008)","journal-title":"Ann. Appl. Probab."},{"key":"1505_CR4","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/j.spa.2004.01.001","volume":"111","author":"B Bouchard","year":"2004","unstructured":"Bouchard, B., Touzi, N.: Discrete-time approximation and monte-carlo simulation of backward stochastic differential equations. Stoch. Proc. Appl. 111, 175\u2013206 (2004)","journal-title":"Stoch. Proc. Appl."},{"key":"1505_CR5","volume-title":"Numerical Analysis, Cengage Learning","author":"RL Burden","year":"2001","unstructured":"Burden, R.L., Faires, J.D.: Numerical Analysis, Cengage Learning, 7th edn. Higher Education Press, Beiging (2001)","edition":"7"},{"key":"1505_CR6","doi-asserted-by":"publisher","DOI":"10.1002\/9780470753767","volume-title":"Numerical Methods for Ordinary Differential Equations","author":"JC Butcher","year":"2008","unstructured":"Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. John Wiley, Chichester, UK (2008)"},{"key":"1505_CR7","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1214\/13-AAP932","volume":"24","author":"D Crisan","year":"2014","unstructured":"Crisan, D., Chassagneux, J.F.: Runge-kutta schemes for backward stochastic differential equations. Ann. Appl. Probab. 24, 679\u2013720 (2014)","journal-title":"Ann. Appl. Probab."},{"issue":"1","key":"1505_CR8","doi-asserted-by":"publisher","first-page":"534","DOI":"10.1137\/090765766","volume":"3","author":"D Crisan","year":"2010","unstructured":"Crisan, D., Manolarakis, K.: Solving backward stochastic differential equations using the cubature method: Application to nonlinear pricing. SIAM J. Finan. Math 3(1), 534\u2013571 (2010)","journal-title":"SIAM J. Finan. Math"},{"key":"1505_CR9","first-page":"940","volume":"16","author":"J Cvitanic","year":"2006","unstructured":"Cvitanic, J., Zhang, J.: The steepest descent method for forward-backward sdes. Electron. J. Probab. 16, 940\u2013968 (2006)","journal-title":"Electron. J. Probab."},{"issue":"1","key":"1505_CR10","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1214\/105051605000000674","volume":"16","author":"F Delarue","year":"2006","unstructured":"Delarue, F., Menozzi, S.: A forward-backward stochastic algorithm for quasi-linear pdes. Ann. Appl. Probab. 16(1), 140\u2013184 (2006)","journal-title":"Ann. Appl. Probab."},{"key":"1505_CR11","doi-asserted-by":"publisher","first-page":"940","DOI":"10.1214\/aoap\/1034968235","volume":"6","author":"J Douglas","year":"1996","unstructured":"Douglas, J., Ma, J., Protter, P.: Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Probab. 6, 940\u2013968 (1996)","journal-title":"Ann. Appl. Probab."},{"key":"1505_CR12","doi-asserted-by":"crossref","unstructured":"E., W., Hutzenthaler, M., Jentzen, A., Kruse, T.: On multilevel picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. J. Sci. Comput. 79, 1534\u20131571 (2019)","DOI":"10.1007\/s10915-018-00903-0"},{"issue":"184","key":"1505_CR13","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1090\/S0025-5718-1988-0935077-0","volume":"51","author":"B Fornberg","year":"1988","unstructured":"Fornberg, B.: Generation of finite difference formulas on arbitrarily spaced grids. Math. Comput. 51(184), 699\u2013706 (1988)","journal-title":"Math. Comput."},{"issue":"9","key":"1505_CR14","first-page":"3439","volume":"22","author":"Y Fu","year":"2017","unstructured":"Fu, Y., Zhao, W., Zhou, T.: Efficient spectral sparse grid approximations for solving multi-dimensional forward backward sdes. Discrete Cont. Dyn-B. 22(9), 3439\u20133458 (2017)","journal-title":"Discrete Cont. Dyn-B."},{"key":"1505_CR15","doi-asserted-by":"publisher","first-page":"2172","DOI":"10.1214\/105051605000000412","volume":"15","author":"E Gobet","year":"2005","unstructured":"Gobet, E., Lemor, J.P., Warin, X.: A regression-based monte carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15, 2172\u20132202 (2005)","journal-title":"Ann. Appl. Probab."},{"issue":"34","key":"1505_CR16","doi-asserted-by":"publisher","first-page":"8505","DOI":"10.1073\/pnas.1718942115","volume":"115","author":"J Han","year":"2018","unstructured":"Han, J., Jentzen, A., Weinan, E.: Solving high-dimensional partial differential equations using deep learning. PNAS 115(34), 8505\u20138510 (2018)","journal-title":"PNAS"},{"key":"1505_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s41546-020-00047-w","volume":"5","author":"J Han","year":"2020","unstructured":"Han, J., Long, J.: Convergence of the deep bsde method for coupled fbsdes. PUQR 5, 1\u201333 (2020)","journal-title":"PUQR"},{"key":"1505_CR18","volume-title":"HPC 2019: Advances in high performance computing","author":"L Kapllani","year":"2020","unstructured":"Kapllani, L., Teng, L., Ehrhardt, M.: A multistep scheme to solve backward stochastic differential equations for option pricing on GPUs. In: Dimov, I., Fidanova, S. (eds.) HPC 2019: Advances in high performance computing. Springer, Cham (2020)"},{"key":"1505_CR19","doi-asserted-by":"publisher","first-page":"889","DOI":"10.3150\/bj\/1161614951","volume":"12","author":"J Lemor","year":"2006","unstructured":"Lemor, J., Gobet, E., Warin, X.: Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12, 889\u2013916 (2006)","journal-title":"Bernoulli"},{"key":"1505_CR20","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1016\/S0167-7152(96)00103-4","volume":"32","author":"JP Lepeltier","year":"1997","unstructured":"Lepeltier, J.P., Martin, J.S.: Backward stochastic differential equations with continuous generator. Statist. Probab. Lett. 32, 425\u2013430 (1997)","journal-title":"Statist. Probab. Lett."},{"issue":"3","key":"1505_CR21","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 explicity-a four step scheme. Probab. Theory Related Fields 98(3), 339\u2013359 (1994)","journal-title":"Probab. Theory Related Fields"},{"issue":"5","key":"1505_CR22","doi-asserted-by":"publisher","first-page":"2636","DOI":"10.1137\/06067393X","volume":"46","author":"J Ma","year":"2008","unstructured":"Ma, J., Shen, J., Zhao, Y.: On numerical approximations of forward-backward stochastic differential equations. SIAM J. Numer. Anal. 46(5), 2636\u20132661 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"1505_CR23","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1016\/j.spa.2004.05.010","volume":"115","author":"J Ma","year":"2005","unstructured":"Ma, J., Zhang, J.: Representations and regularities for solutions to bsdes with reflections. Stoch. Proc. Appl. 115, 539\u2013569 (2005)","journal-title":"Stoch. Proc. Appl."},{"key":"1505_CR24","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1137\/040614426","volume":"28","author":"GN Milsetin","year":"2006","unstructured":"Milsetin, G.N., Tretyakov, M.V.: Numerical algorithms for forward-backward stochastic differential equations. SIAM J. Sci. Comput. 28, 561\u2013582 (2006)","journal-title":"SIAM J. Sci. Comput."},{"key":"1505_CR25","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.: Adapted solution of a backward stochastic differential equations. Syst. Control Lett. 14, 55\u201361 (1990)","journal-title":"Syst. Control Lett."},{"key":"1505_CR26","first-page":"200","volume":"176","author":"E Pardoux","year":"1992","unstructured":"Pardoux, E., Peng, S.: Backward stochastic differential equation and quasilinear parabolic partial differential equations. Lectures Notes in CSI. 176, 200\u2013217 (1992)","journal-title":"Lectures Notes in CSI."},{"issue":"1\u20132","key":"1505_CR27","first-page":"61","volume":"37","author":"S Peng","year":"1991","unstructured":"Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastic Rep. 37(1\u20132), 61\u201374 (1991)","journal-title":"Stochastics Stochastic Rep."},{"key":"1505_CR28","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1137\/S0363012996313549","volume":"37","author":"S Peng","year":"1999","unstructured":"Peng, S., Wu, Z.: Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37, 825\u2013843 (1999)","journal-title":"SIAM J. Control Optim."},{"key":"1505_CR29","unstructured":"Raissi, M.: Forward-backward stochastic neural networks: Deep learning of high-dimensional partial differential equations. arXiv:1804.07010v1, available on webpage at arXiv:1804.07010v1.pdf (2018)"},{"issue":"2","key":"1505_CR30","doi-asserted-by":"publisher","first-page":"A859","DOI":"10.1137\/130913183","volume":"37","author":"MJ Ruijter","year":"2015","unstructured":"Ruijter, M.J., Oosterlee, C.W.: A fourier cosine method for an efficient computation of solutions to bsdes. SIAM J. Sci. Comput. 37(2), A859\u2013A889 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"1505_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral Methods: Algorithms, Analysis and Applications.","author":"J Shen","year":"2011","unstructured":"Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer-Verlag, Berlin (2011)"},{"key":"1505_CR32","unstructured":"Teng, L.: A review of tree-based approaches to solve forward-backward stochastic differential equations. To appear in J. Comput. J. Comput. Finance (forthcoming) (2019)"},{"key":"1505_CR33","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/j.apnum.2019.09.016","volume":"150","author":"L Teng","year":"2020","unstructured":"Teng, L., Lapitckii, A., G\u00fcnther, M.: A multi-step scheme based on cubic spline for solving backward stochastic differential equations. Appl. Numer. Math. 150, 117\u2013138 (2020)","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"1505_CR34","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(3), 221\u2013248 (2013)","journal-title":"J. Comput. Math."},{"key":"1505_CR35","unstructured":"Zhang, J.: Some fine properties of backward stochastic differential equations. Ph.D. thesis, Purdue University, West Lafayette, IN (2001)"},{"key":"1505_CR36","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."},{"issue":"4","key":"1505_CR37","doi-asserted-by":"publisher","first-page":"1563","DOI":"10.1137\/05063341X","volume":"28","author":"W Zhao","year":"2006","unstructured":"Zhao, W., Chen, L., Peng, S.: A new kind of accurate numerical method for backward stochastic differential equations. SIAM J. Sci. Comput. 28(4), 1563\u20131581 (2006)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"1505_CR38","doi-asserted-by":"publisher","first-page":"A1731","DOI":"10.1137\/130941274","volume":"36","author":"W Zhao","year":"2014","unstructured":"Zhao, W., Fu, Y., Zhou, T.: New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations. SIAM J. Sci. Comput. 36(4), A1731\u2013A1751 (2014)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"1505_CR39","first-page":"876","volume":"10","author":"W Zhao","year":"2013","unstructured":"Zhao, W., Li, Y., Ju, L.: Error estimates of the crank-nicolson scheme for solving backward stochastic differential equations. Int. J. Numer. Anal. Mode. l 10(4), 876\u2013898 (2013)","journal-title":"Int. J. Numer. Anal. Mode. l"},{"issue":"5","key":"1505_CR40","doi-asserted-by":"publisher","first-page":"1585","DOI":"10.3934\/dcdsb.2012.17.1585","volume":"17","author":"W Zhao","year":"2012","unstructured":"Zhao, W., Li, Y., Zhang, G.: A generalized $$\\theta $$-scheme for solving backward stochastic differential equations. Discrete Cont. Dyn-B. 17(5), 1585\u20131603 (2012)","journal-title":"Discrete Cont. Dyn-B."},{"key":"1505_CR41","first-page":"905","volume":"12","author":"W Zhao","year":"2009","unstructured":"Zhao, W., Wang, J., Peng, S.: Error estimates of the theta-scheme for backward stochastic differential equations. Discrete Contin. Dyn. Syst. Ser. B 12, 905\u2013924 (2009)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"1505_CR42","doi-asserted-by":"publisher","first-page":"1369","DOI":"10.1137\/09076979X","volume":"48","author":"W Zhao","year":"2010","unstructured":"Zhao, W., Zhang, G., Ju, L.: A stable multistep scheme for solving backward stochastic differential equations. SIAM J. Numer. Anal. 48, 1369\u20131394 (2010)","journal-title":"SIAM J. Numer. Anal."},{"key":"1505_CR43","doi-asserted-by":"publisher","first-page":"618","DOI":"10.4208\/cicp.280113.190813a","volume":"15","author":"W Zhao","year":"2014","unstructured":"Zhao, W., Zhang, W., Ju, L.: A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations. Commun. Comput. Phys. 15, 618\u2013646 (2014)","journal-title":"Commun. Comput. Phys."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01505-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-021-01505-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01505-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,24]],"date-time":"2021-05-24T18:29:04Z","timestamp":1621880944000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-021-01505-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,29]]},"references-count":43,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["1505"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01505-z","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,29]]},"assertion":[{"value":"12 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 March 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 April 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2021","order":4,"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 that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"81"}}