{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T10:54:56Z","timestamp":1776336896632,"version":"3.51.2"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T00:00:00Z","timestamp":1540339200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11671343"],"award-info":[{"award-number":["11671343"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Hunan Province Innovation Foundation for Postgraduate","award":["CX2016B250"],"award-info":[{"award-number":["CX2016B250"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2019,4]]},"DOI":"10.1007\/s10444-018-9638-0","type":"journal-article","created":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T14:14:31Z","timestamp":1540390471000},"page":"813-846","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Asymptotically optimal approximation of some stochastic integrals and its applications to the strong second-order methods"],"prefix":"10.1007","volume":"45","author":[{"given":"Xiao","family":"Tang","sequence":"first","affiliation":[]},{"given":"Aiguo","family":"Xiao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,10,24]]},"reference":[{"issue":"3","key":"9638_CR1","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1023\/A:1022363612387","volume":"40","author":"L Brugnano","year":"2000","unstructured":"Brugnano, L., Burrage, K., Burrage, P.M.: Adams-type methods for the numerical solution of stochastic ordinary differential equations. BIT Numer. Math. 40(3), 451\u2013470 (2000)","journal-title":"BIT Numer. Math."},{"issue":"2","key":"9638_CR2","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1137\/040602857","volume":"44","author":"E Buckwar","year":"2006","unstructured":"Buckwar, E., Winkler, R.: Multistep methods for SDEs and their application to problem with small noise. SIAM J. Numer. Anal. 44(2), 779\u2013803 (2006)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"9638_CR3","doi-asserted-by":"publisher","first-page":"912","DOI":"10.1016\/j.cam.2006.03.038","volume":"205","author":"E Buckwar","year":"2007","unstructured":"Buckwar, E., Winkler, R.: Improved linear multi-step methods for stochastic ordinary differential equations. J. Comput. Appl. Math. 205(2), 912\u2013922 (2007)","journal-title":"J. Comput. Appl. Math."},{"key":"9638_CR4","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/S0168-9274(96)00027-X","volume":"22","author":"K Burrage","year":"1996","unstructured":"Burrage, K., Burrage, P.M.: High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations. Appl. Numer. Math. 22, 81\u2013101 (1996)","journal-title":"Appl. Numer. Math."},{"key":"9638_CR5","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/S0168-9274(98)00042-7","volume":"28","author":"K Burrage","year":"1998","unstructured":"Burrage, K., Burrage, P.M.: General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems. Appl. Numer. Math. 28, 161\u2013177 (1998)","journal-title":"Appl. Numer. Math."},{"issue":"5","key":"9638_CR6","doi-asserted-by":"publisher","first-page":"1626","DOI":"10.1137\/S0036142999363206","volume":"38","author":"K Burrage","year":"2000","unstructured":"Burrage, K., Burrage, P.M.: Order conditions of stochastic Runge-Kutta methods by B-series. SIAM J. Numer. Anal. 38(5), 1626\u20131646 (2000)","journal-title":"SIAM J. Numer. Anal."},{"key":"9638_CR7","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1098\/rspa.2003.1247","volume":"460","author":"K Burrage","year":"2004","unstructured":"Burrage, K., Burrage, P.M., Tian, T.: Numerical methods for strong solutions of stochastic differential equations: An overview. Proc. R. Soc. Lond. A 460, 373\u2013402 (2004)","journal-title":"Proc. R. Soc. Lond. A"},{"issue":"6","key":"9638_CR8","doi-asserted-by":"publisher","first-page":"821","DOI":"10.4208\/aamm.12-12S11","volume":"4","author":"W Cao","year":"2012","unstructured":"Cao, W., Zhang, Z.: Simulations of two-step Maruyama methods for nonlinear stochastic delay differential equations. Adv. Appl. Math. Mech. 4(6), 821\u2013832 (2012)","journal-title":"Adv. Appl. Math. Mech."},{"issue":"3","key":"9638_CR9","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1007\/s10543-010-0276-2","volume":"50","author":"K Debrabant","year":"2010","unstructured":"Debrabant, K.: Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise. BIT Numer. Math. 50(3), 541\u2013558 (2010)","journal-title":"BIT Numer. Math."},{"issue":"1","key":"9638_CR10","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1137\/070704307","volume":"47","author":"K Debrabant","year":"2008","unstructured":"Debrabant, K., Kv\u00e6r\u00f8n, A.: B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J. Numer. Anal. 47(1), 181\u2013203 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"9638_CR11","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1007\/s10543-010-0272-6","volume":"50","author":"H la Cruz Cancino De","year":"2010","unstructured":"De la Cruz Cancino, H., Biscay, R.J., Jimenez, J.C., Carbonell, F., Ozaki, T.: High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise. BIT Numer. Math. 50, 509\u2013539 (2010)","journal-title":"BIT Numer. Math."},{"issue":"5","key":"9638_CR12","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1080\/07362990701540592","volume":"25","author":"AS Dickinson","year":"2007","unstructured":"Dickinson, A.S.: Optimal approximation of the second iterated integral of Brownian motion. Stoch. Anal. Appl. 25(5), 1109\u20131128 (2007)","journal-title":"Stoch. Anal. Appl."},{"key":"9638_CR13","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1016\/0021-9991(84)90086-X","volume":"56","author":"AL Fogelson","year":"1984","unstructured":"Fogelson, A.L.: A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting. J. Comput. Phys. 56, 111\u2013134 (1984)","journal-title":"J. Comput. Phys."},{"key":"9638_CR14","first-page":"201","volume":"8","author":"S Gan","year":"2011","unstructured":"Gan, S., Henri, S., Zhang, H.: Mean square convergence of stochastic \ud835\udf03-methods for nonlinear neutral stochastic differential delay equations. Int. J. Numer. Anal. Mod. 8, 201\u2013213 (2011)","journal-title":"Int. J. Numer. Anal. Mod."},{"issue":"5","key":"9638_CR15","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1137\/0324060","volume":"24","author":"S Geman","year":"1986","unstructured":"Geman, S., Hwang, C.R.: Diffusions for global optimization. SIAM J. Control Optim. 24(5), 1031\u20131043 (1986)","journal-title":"SIAM J. Control Optim."},{"key":"9638_CR16","doi-asserted-by":"publisher","DOI":"10.1142\/5949","volume-title":"The noisy oscillator","author":"M Gitterman","year":"2005","unstructured":"Gitterman, M.: The noisy oscillator. World Scientific, Singapore (2005)"},{"key":"9638_CR17","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1137\/S0036144500378302","volume":"43","author":"DJ Higham","year":"2001","unstructured":"Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525\u2013546 (2001)","journal-title":"SIAM Rev."},{"key":"9638_CR18","doi-asserted-by":"publisher","first-page":"1041","DOI":"10.1137\/S0036142901389530","volume":"40","author":"DJ Higham","year":"2002","unstructured":"Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041\u20131063 (2002)","journal-title":"SIAM J. Numer. Anal."},{"key":"9638_CR19","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.apnum.2014.08.003","volume":"87","author":"J Hong","year":"2015","unstructured":"Hong, J., Xu, D., Wang, P.: Preservation of quadratic invariants of stochastic differential equations via Runge-Kutta methods. Appl. Numer. Math. 87, 38\u201352 (2015)","journal-title":"Appl. Numer. Math."},{"key":"9638_CR20","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511526169","volume-title":"Gaussian Hilbert spaces","author":"S Janson","year":"1997","unstructured":"Janson, S.: Gaussian Hilbert spaces. Cambridge University Press, Cambridge (1997)"},{"key":"9638_CR21","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1016\/j.cam.2014.10.021","volume":"279","author":"JC Jim\u00e9nez","year":"2015","unstructured":"Jim\u00e9nez, J.C., Carbonell, F.: Convergence rate of weak local linearization schemes for stochastic differential equations with additive noise. J. Comput. Appl. Math. 279, 106\u2013122 (2015)","journal-title":"J. Comput. Appl. Math."},{"key":"9638_CR22","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1007\/s10543-011-0360-2","volume":"52","author":"JC Jimenez","year":"2012","unstructured":"Jimenez, J.C., De la Cruz Cancino, H.: Convergence rate of strong local linearization schemes for stochastic differential equations with additive noise. BIT Numer. Math. 52, 357\u2013382 (2012)","journal-title":"BIT Numer. Math."},{"key":"9638_CR23","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1007\/s11075-017-0440-8","volume":"79","author":"M Kamrani","year":"2018","unstructured":"Kamrani, M., Jamshidi, N.: Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation. Numer. Algor. 79, 357\u2013374 (2018)","journal-title":"Numer. Algor."},{"key":"9638_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12616-5","volume-title":"Numerical solution of stochastic differential equations","author":"PE Kloeden","year":"1992","unstructured":"Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. Springer, Berlin (1992)"},{"issue":"1","key":"9638_CR25","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1016\/j.cam.2006.06.006","volume":"206","author":"Y Komori","year":"2007","unstructured":"Komori, Y.: Weak second-order stochastic Runge-Kutta methods for non-commutative stochastic differential equations. J. Comput. Appl. Math. 206(1), 158\u2013173 (2007)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"9638_CR26","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/BF02510172","volume":"37","author":"Y Komori","year":"1997","unstructured":"Komori, Y., Mitsui, T., Sugiura, H.: Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations. BIT Numer. Math. 37(1), 43\u201366 (1997)","journal-title":"BIT Numer. Math."},{"key":"9638_CR27","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.amc.2015.04.003","volume":"261","author":"X Li","year":"2015","unstructured":"Li, X., Cao, W.: On mean-square stability of two-step Maruyama methods for nonlinear neutral stochastic delay differential equations. Appl. Math. Comput. 261, 373\u2013381 (2015)","journal-title":"Appl. Math. Comput."},{"key":"9638_CR28","volume-title":"Stochastic differential equations and applications","author":"X Mao","year":"1997","unstructured":"Mao, X.: Stochastic differential equations and applications. Horword, Chichester (1997)"},{"key":"9638_CR29","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1016\/j.cam.2015.06.002","volume":"290","author":"X Mao","year":"2015","unstructured":"Mao, X.: The truncated Euler-Maruyama method for stochastic differential equations. J. Comput. Appl. Math. 290, 370\u2013384 (2015)","journal-title":"J. Comput. Appl. Math."},{"key":"9638_CR30","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1016\/j.cam.2015.09.035","volume":"296","author":"X Mao","year":"2016","unstructured":"Mao, X.: Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations. J. Comput. Appl. Math. 296, 362\u2013375 (2016)","journal-title":"J. Comput. Appl. Math."},{"key":"9638_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-8455-5","volume-title":"Numerical integration of stochastic differential equations","author":"GN Milstein","year":"1995","unstructured":"Milstein, G.N.: Numerical integration of stochastic differential equations. Kluwer, Dordrecht (1995)"},{"key":"9638_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-10063-9","volume-title":"Stochastic numerics for mathematical physics","author":"GN Milstein","year":"2004","unstructured":"Milstein, G.N., Tretyakov, M.V.: Stochastic numerics for mathematical physics. Springer, Berlin (2004)"},{"key":"9638_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2437-0","volume-title":"The Malliavin calculus and related topics","author":"D Nualart","year":"1995","unstructured":"Nualart, D.: The Malliavin calculus and related topics. Springer-Verlag, New York (1995)"},{"key":"9638_CR34","doi-asserted-by":"publisher","first-page":"896","DOI":"10.1002\/hbm.20230","volume":"27","author":"JJ Riera","year":"2006","unstructured":"Riera, J.J., Wan, X., Jimenez, J.C., Kawashima, R.: Nonlinear local electrovascular coupling. I: a theoretical model. Hum. Brain Mapp. 27, 896\u2013914 (2006)","journal-title":"Hum. Brain Mapp."},{"issue":"1","key":"9638_CR35","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1080\/07362990500397699","volume":"24","author":"A R\u00f6\u00dfler","year":"2006","unstructured":"R\u00f6\u00dfler, A.: Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations. Stochastic. Anal. Appl. 24(1), 97\u2013134 (2006)","journal-title":"Stochastic. Anal. Appl."},{"issue":"3","key":"9638_CR36","doi-asserted-by":"publisher","first-page":"1713","DOI":"10.1137\/060673308","volume":"47","author":"A R\u00f6\u00dfler","year":"2009","unstructured":"R\u00f6\u00dfler, A.: Second order Runge-Kutta methods for It\u00f4 stochastic differential equations. SIAM J. Numer. Anal. 47(3), 1713\u20131738 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"9638_CR37","doi-asserted-by":"publisher","first-page":"922","DOI":"10.1137\/09076636X","volume":"48","author":"A R\u00f6\u00dfler","year":"2010","unstructured":"R\u00f6\u00dfler, A.: Runge-Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. Numer. Anal. 48(3), 922\u2013952 (2010)","journal-title":"SIAM J. Numer. Anal."},{"key":"9638_CR38","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1007\/s10543-016-0618-9","volume":"57","author":"X Tang","year":"2017","unstructured":"Tang, X., Xiao, A.: Efficient weak second-order stochastic Runge-Kutta methods for It\u00f4 stochastic differential equations. BIT Numer. Math. 57, 241\u2013260 (2017)","journal-title":"BIT Numer. Math."},{"issue":"2","key":"9638_CR39","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/s11075-012-9568-8","volume":"62","author":"X Wang","year":"2013","unstructured":"Wang, X., Gan, S.: A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise. Numer. Algor. 62 (2), 193\u2013223 (2013)","journal-title":"Numer. Algor."},{"key":"9638_CR40","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s11075-015-0044-0","volume":"72","author":"A Xiao","year":"2016","unstructured":"Xiao, A., Tang, X.: High strong order stochastic Runge-Kutta methods for Stratonovich stochastic differential equations with scalar noise. Numer. Algor. 72, 259\u2013296 (2016)","journal-title":"Numer. Algor."},{"key":"9638_CR41","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1016\/j.cam.2017.04.050","volume":"325","author":"W Zhou","year":"2017","unstructured":"Zhou, W., Zhang, J., Hong, J., Song, S.: Stochastic symplectic Runge-Kutta methods for the strong approximation of Hamiltonian systems with additive noise. J. Comput. Appl. Math. 325, 134\u2013148 (2017)","journal-title":"J. Comput. Appl. Math."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-018-9638-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-018-9638-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-018-9638-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,4]],"date-time":"2022-09-04T15:22:52Z","timestamp":1662304972000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-018-9638-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,24]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,4]]}},"alternative-id":["9638"],"URL":"https:\/\/doi.org\/10.1007\/s10444-018-9638-0","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,24]]},"assertion":[{"value":"8 January 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 September 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 October 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}