{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T23:59:21Z","timestamp":1772668761002,"version":"3.50.1"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,9,3]],"date-time":"2024-09-03T00:00:00Z","timestamp":1725321600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,9,3]],"date-time":"2024-09-03T00:00:00Z","timestamp":1725321600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s11075-024-01925-6","type":"journal-article","created":{"date-parts":[[2024,9,3]],"date-time":"2024-09-03T06:02:35Z","timestamp":1725343355000},"page":"1601-1622","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The fast Euler-Maruyama method for solving multiterm Caputo fractional stochastic delay integro-differential equations"],"prefix":"10.1007","volume":"99","author":[{"given":"Huijiao","family":"Guo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jin","family":"Huang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yi","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xueli","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,9,3]]},"reference":[{"key":"1925_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2021.113825","volume":"406","author":"S Behera","year":"2022","unstructured":"Behera, S., Saha Ray, S.: An efficient numerical method based on Euler wavelets for solving fractional order pantograph Volterra delay-integro-differential equations. J. Comput. Appl. Math. 406, 113825 (2022)","journal-title":"J. Comput. Appl. Math."},{"key":"1925_CR2","doi-asserted-by":"crossref","unstructured":"Evans, L.C.: An introduction to stochastic differential equations. (2013)","DOI":"10.1090\/mbk\/082"},{"key":"1925_CR3","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1016\/j.cam.2019.02.002","volume":"356","author":"X Dai","year":"2019","unstructured":"Dai, X., Bu, W., Xiao, A.: Well-posedness and EM approximations for non-lipschitz stochastic fractional integro-differential equations. J. Comput. Appl. Math. 356, 377\u2013390 (2019)","journal-title":"J. Comput. Appl. Math."},{"key":"1925_CR4","doi-asserted-by":"crossref","unstructured":"Doan, T.S., Huong, P.T., Kloeden, P.E., Vu, A.M.: Euler-Maruyama scheme for Caputo stochastic fractional differential equations. J. Comput. Appl. Math. 380 (2020)","DOI":"10.1016\/j.cam.2020.112989"},{"key":"1925_CR5","doi-asserted-by":"crossref","unstructured":"Dong, J., Du, N., Yang, Z.: A distributed-order fractional stochastic differential equation driven by L\u00e9vy noise: Existence, uniqueness, and a fast EM scheme. Chaos: An Interdisciplinary J. Nonlinear Sci. 33(2), 023109 (2023)","DOI":"10.1063\/5.0135471"},{"key":"1925_CR6","doi-asserted-by":"publisher","first-page":"A2510","DOI":"10.1137\/18M1230153","volume":"41","author":"L Guo","year":"2018","unstructured":"Guo, L., Zeng, F., Turner, I., Burrage, K., Karniadakis, G.E.: Efficient multistep methods for tempered fractional calculus: Algorithms and simulations. SIAM J. Sci. Comput. 41, A2510\u2013A2535 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"1925_CR7","doi-asserted-by":"publisher","first-page":"679","DOI":"10.4208\/nmtma.OA-2022-0023","volume":"15","author":"Y Huang","year":"2022","unstructured":"Huang, Y., Li, Q., Li, R., Zeng, F., Guo, L.: A unified fast memory-saving time-stepping method for fractional operators and its applications. Numerical Mathematics-theory Methods Appl. 15(3), 679\u2013714 (2022)","journal-title":"Numerical Mathematics-theory Methods Appl."},{"key":"1925_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2022.109768","volume":"195","author":"PT Huong","year":"2023","unstructured":"Huong, P.T., The, N.T.: Well-posedness and regularity for solutions of Caputo stochastic fractional delay differential equations. Statistics Probability Lett. 195, 109768 (2023)","journal-title":"Statistics Probability Lett."},{"issue":"3","key":"1925_CR9","doi-asserted-by":"publisher","first-page":"650","DOI":"10.4208\/cicp.OA-2016-0136","volume":"21","author":"S Jiang","year":"2017","unstructured":"Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21(3), 650\u2013678 (2017)","journal-title":"Commun. Comput. Phys."},{"key":"1925_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2021.107515","volume":"123","author":"N Kopteva","year":"2022","unstructured":"Kopteva, N.: Pointwise-in-time a posteriori error control for time-fractional parabolic equations. Appl. Math. Lett. 123, 107515 (2022)","journal-title":"Appl. Math. Lett."},{"key":"1925_CR11","doi-asserted-by":"crossref","unstructured":"Kopteva, N., Stynes, M.: A posteriori error analysis for variable-coefficient multiterm time-fractional subdiffusion equations. J. Scientific Comput. 92(2) (2022)","DOI":"10.1007\/s10915-022-01936-2"},{"issue":"4","key":"1925_CR12","doi-asserted-by":"publisher","first-page":"2433","DOI":"10.1007\/s11075-022-01397-6","volume":"92","author":"M Li","year":"2023","unstructured":"Li, M., Dai, X., Huang, C.: Fast Euler-Maruyama method for weakly singular stochastic Volterra integral equations with variable exponent. Numerical Algorithms 92(4), 2433\u20132455 (2023)","journal-title":"Numerical Algorithms"},{"key":"1925_CR13","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s12190-011-0510-3","volume":"39","author":"Q Li","year":"2011","unstructured":"Li, Q., Gan, S.: Mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations. J. Appl. Math. Comput. 39, 69\u201387 (2011)","journal-title":"J. Appl. Math. Comput."},{"key":"1925_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2021.126220","volume":"581","author":"LS Lima","year":"2021","unstructured":"Lima, L.S., Melgaco, J.H.C.: Dynamics of stocks prices based in the black and scholes equation and nonlinear stochastic differentials equations. Physica A: Statistical Mechanics Appl. 581, 126220 (2021)","journal-title":"Physica A: Statistical Mechanics Appl."},{"key":"1925_CR15","doi-asserted-by":"crossref","unstructured":"McLean, W.: Exponential sum approximations for $$t^{-\\beta }$$. arXiv: Numerical Analysis pages 911\u2013930 (2016)","DOI":"10.1007\/978-3-319-72456-0_40"},{"key":"1925_CR16","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/j.enganabound.2018.05.006","volume":"100","author":"F Mirzaee","year":"2019","unstructured":"Mirzaee, F., Samadyar, N.: On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions. Eng. Anal. Boundary Elements 100, 246\u2013255 (2019)","journal-title":"Eng. Anal. Boundary Elements"},{"key":"1925_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2020.105475","volume":"92","author":"BP Moghaddam","year":"2021","unstructured":"Moghaddam, B.P., Mostaghim, Z.S., Pantelous, A.A., Tenreiro Machado, J.A.: An integro quadratic spline-based scheme for solving nonlinear fractional stochastic differential equations with constant time delay. Commun. Nonlinear Sci. Numerical Simulation 92, 105475 (2021)","journal-title":"Commun. Nonlinear Sci. Numerical Simulation"},{"key":"1925_CR18","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1016\/j.jcp.2015.05.051","volume":"298","author":"F Mohammadi","year":"2015","unstructured":"Mohammadi, F.: A wavelet-based computational method for solving stochastic It$$\\widehat{o}$$-Volterra integral equations. J. Comput. Phys. 298, 254\u2013265 (2015)","journal-title":"J. Comput. Phys."},{"key":"1925_CR19","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1137\/130932132","volume":"56","author":"LN Trefethen","year":"2014","unstructured":"Trefethen, L.N., Weideman, A.: The exponentially convergent trapezoidal rule. SIAM Rev. 56, 385\u2013458 (2014)","journal-title":"SIAM Rev."},{"issue":"2","key":"1925_CR20","doi-asserted-by":"publisher","first-page":"1075","DOI":"10.1016\/j.jmaa.2006.05.061","volume":"328","author":"H Ye","year":"2007","unstructured":"Ye, H., Gao, J., Ding, Y.: A generalized gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075\u20131081 (2007)","journal-title":"J. Math. Anal. Appl."},{"key":"1925_CR21","doi-asserted-by":"crossref","unstructured":"Zhang, J., Lv, J., Huang, J., Tang, Y.: A fast Euler-Maruyama method for Riemann-Liouville stochastic fractional nonlinear differential equations. Physica D-Nolinear Phenomena 446 (2023)","DOI":"10.1016\/j.physd.2023.133685"},{"key":"1925_CR22","doi-asserted-by":"crossref","unstructured":"Yuan, H.: Convergence and stability of exponential integrators for semi-linear stochastic variable delay integro-differential equations. Int. J. Comput. Math. 98(5) (2020)","DOI":"10.1080\/00207160.2020.1792452"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01925-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01925-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01925-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T08:43:31Z","timestamp":1750149811000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01925-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,3]]},"references-count":22,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["1925"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01925-6","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,3]]},"assertion":[{"value":"12 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 August 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 September 2024","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":"Competing interests"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}}]}}