{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T10:15:13Z","timestamp":1783678513043,"version":"3.55.0"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T00:00:00Z","timestamp":1781049600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T00:00:00Z","timestamp":1781049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271368"],"award-info":[{"award-number":["12271368"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013105","name":"Shanghai Rising-Star Program","doi-asserted-by":"publisher","award":["22QA1406900"],"award-info":[{"award-number":["22QA1406900"]}],"id":[{"id":"10.13039\/501100013105","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1007\/s10915-026-03354-0","type":"journal-article","created":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T06:49:44Z","timestamp":1781074184000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Numerical Approximations to Invariant Measures of Hybrid Stochastic Differential Equations with Superlinear Coefficients Via the Backward Euler-Maruyama Method"],"prefix":"10.1007","volume":"108","author":[{"given":"Wei","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jie","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,10]]},"reference":[{"key":"3354_CR1","doi-asserted-by":"publisher","unstructured":"Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006). https:\/\/doi.org\/10.1142\/p473","DOI":"10.1142\/p473"},{"key":"3354_CR2","doi-asserted-by":"publisher","unstructured":"Yin, G., Zhu, C.: Hybrid Switching Diffusions. Stochastic Modelling and Applied Probability, vol. 63. Springer, New York (2010). https:\/\/doi.org\/10.1007\/978-1-4419-1105-6","DOI":"10.1007\/978-1-4419-1105-6"},{"issue":"1","key":"3354_CR3","doi-asserted-by":"publisher","first-page":"725","DOI":"10.1137\/15M1024512","volume":"48","author":"J Bao","year":"2016","unstructured":"Bao, J., Shao, J.: Permanence and extinction of regime-switching predator-prey models. SIAM J. Math. Anal. 48(1), 725\u2013739 (2016). https:\/\/doi.org\/10.1137\/15M1024512","journal-title":"SIAM J. Math. Anal."},{"issue":"2","key":"3354_CR4","doi-asserted-by":"publisher","first-page":"496","DOI":"10.1016\/j.jmaa.2012.05.029","volume":"394","author":"A Gray","year":"2012","unstructured":"Gray, A., Greenhalgh, D., Mao, X., Pan, J.: The SIS epidemic model with Markovian switching. J. Math. Anal. Appl. 394(2), 496\u2013516 (2012). https:\/\/doi.org\/10.1016\/j.jmaa.2012.05.029","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"3354_CR5","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1142\/S0219024910005863","volume":"13","author":"R Liu","year":"2010","unstructured":"Liu, R.: Regime-switching recombining tree for option pricing. Int. J. Theor. Appl. Finance 13(3), 479\u2013499 (2010). https:\/\/doi.org\/10.1142\/S0219024910005863","journal-title":"Int. J. Theor. Appl. Finance"},{"key":"3354_CR6","doi-asserted-by":"publisher","first-page":"482","DOI":"10.1016\/j.apm.2019.10.010","volume":"78","author":"Y Deng","year":"2020","unstructured":"Deng, Y., Liu, M.: Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations. Appl. Math. Model. 78, 482\u2013504 (2020). https:\/\/doi.org\/10.1016\/j.apm.2019.10.010","journal-title":"Appl. Math. Model."},{"issue":"2","key":"3354_CR7","doi-asserted-by":"publisher","first-page":"592","DOI":"10.1137\/060658138","volume":"45","author":"DJ Higham","year":"2007","unstructured":"Higham, D.J., Mao, X., Yuan, C.: Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J. Numer. Anal. 45(2), 592\u2013609 (2007). https:\/\/doi.org\/10.1137\/060658138","journal-title":"SIAM J. Numer. Anal."},{"issue":"2130","key":"3354_CR8","doi-asserted-by":"publisher","first-page":"1563","DOI":"10.1098\/rspa.2010.0348","volume":"467","author":"M Hutzenthaler","year":"2011","unstructured":"Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong and weak divergence in finite time of Euler\u2019s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467(2130), 1563\u20131576 (2011). https:\/\/doi.org\/10.1098\/rspa.2010.0348","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"3354_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.112917","volume":"377","author":"C Kumar","year":"2020","unstructured":"Kumar, C., Kumar, T.: On explicit tamed Milstein-type scheme for stochastic differential equation with Markovian switching. J. Comput. Appl. Math. 377, 112917 (2020). https:\/\/doi.org\/10.1016\/j.cam.2020.112917","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"3354_CR10","doi-asserted-by":"publisher","first-page":"936","DOI":"10.1016\/j.cam.2006.01.052","volume":"205","author":"X Mao","year":"2007","unstructured":"Mao, X., Yuan, C., Yin, G.: Approximations of Euler-Maruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions. J. Comput. Appl. Math. 205(2), 936\u2013948 (2007). https:\/\/doi.org\/10.1016\/j.cam.2006.01.052","journal-title":"J. Comput. Appl. Math."},{"key":"3354_CR11","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1016\/j.nahs.2018.04.003","volume":"30","author":"DT Nguyen","year":"2018","unstructured":"Nguyen, D.T., Nguyen, S.L., Hoang, T.A., Yin, G.: Tamed-Euler method for hybrid stochastic differential equations with Markovian switching. Nonlinear Anal. Hybrid Syst 30, 14\u201330 (2018). https:\/\/doi.org\/10.1016\/j.nahs.2018.04.003","journal-title":"Nonlinear Anal. Hybrid Syst"},{"issue":"2","key":"3354_CR12","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1137\/16M1084730","volume":"55","author":"SL Nguyen","year":"2017","unstructured":"Nguyen, S.L., Hoang, T.A., Nguyen, D.T., Yin, G.: Milstein-type procedures for numerical solutions of stochastic differential equations with Markovian switching. SIAM J. Numer. Anal. 55(2), 953\u2013979 (2017). https:\/\/doi.org\/10.1137\/16M1084730","journal-title":"SIAM J. Numer. Anal."},{"key":"3354_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2023.115197","volume":"429","author":"Y Zhang","year":"2023","unstructured":"Zhang, Y., Song, M., Liu, M.: Strong convergence of the tamed Euler method for nonlinear hybrid stochastic differential equations with piecewise continuous arguments. J. Comput. Appl. Math. 429, 115197 (2023). https:\/\/doi.org\/10.1016\/j.cam.2023.115197","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"3354_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cam.2004.03.016","volume":"174","author":"X Mao","year":"2005","unstructured":"Mao, X., Yuan, C., Yin, G.: Numerical method for stationary distribution of stochastic differential equations with Markovian switching. J. Comput. Appl. Math. 174(1), 1\u201327 (2005). https:\/\/doi.org\/10.1016\/j.cam.2004.03.016","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"3354_CR15","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1080\/10236190412331314150","volume":"11","author":"C Yuan","year":"2005","unstructured":"Yuan, C., Mao, X.: Stationary distributions of Euler-Maruyama-type stochastic difference equations with Markovian switching and their convergence. J. Difference Equ. Appl. 11(1), 29\u201348 (2005). https:\/\/doi.org\/10.1080\/10236190412331314150","journal-title":"J. Difference Equ. Appl."},{"issue":"4","key":"3354_CR16","doi-asserted-by":"publisher","first-page":"707","DOI":"10.1007\/s11118-015-9526-x","volume":"44","author":"J Bao","year":"2016","unstructured":"Bao, J., Shao, J., Yuan, C.: Approximation of invariant measures for regime-switching diffusions. Potential Anal. 44(4), 707\u2013727 (2016). https:\/\/doi.org\/10.1007\/s11118-015-9526-x","journal-title":"Potential Anal."},{"issue":"5","key":"3354_CR17","doi-asserted-by":"publisher","first-page":"3215","DOI":"10.1137\/18M116678X","volume":"56","author":"J Shao","year":"2018","unstructured":"Shao, J.: Invariant measures and Euler-Maruyama\u2019s approximations of state-dependent regime-switching diffusions. SIAM J. Control. Optim. 56(5), 3215\u20133238 (2018). https:\/\/doi.org\/10.1137\/18M116678X","journal-title":"SIAM J. Control. Optim."},{"key":"3354_CR18","doi-asserted-by":"publisher","unstructured":"Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II, 2nd edn. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (1996). https:\/\/doi.org\/10.1007\/978-3-642-05221-7","DOI":"10.1007\/978-3-642-05221-7"},{"issue":"3","key":"3354_CR19","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1093\/imamat\/hxr016","volume":"76","author":"DJ Higham","year":"2011","unstructured":"Higham, D.J.: Stochastic ordinary differential equations in applied and computational mathematics. IMA J. Appl. Math. 76(3), 449\u2013474 (2011). https:\/\/doi.org\/10.1093\/imamat\/hxr016","journal-title":"IMA J. Appl. Math."},{"key":"3354_CR20","doi-asserted-by":"publisher","unstructured":"Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics, 2nd edn. Scientific Computation. Springer, Cham (2021). https:\/\/doi.org\/10.1007\/978-3-030-82040-4","DOI":"10.1007\/978-3-030-82040-4"},{"key":"3354_CR21","doi-asserted-by":"publisher","unstructured":"Higham, D.J., Kloeden, P.E.: An Introduction to the Numerical Simulation of Stochastic Differential Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2021). https:\/\/doi.org\/10.1137\/1.9781611976434","DOI":"10.1137\/1.9781611976434"},{"issue":"7","key":"3354_CR22","doi-asserted-by":"publisher","first-page":"2921","DOI":"10.1016\/j.jde.2018.04.052","volume":"265","author":"H Yang","year":"2018","unstructured":"Yang, H., Li, X.: Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons. J. Differential Equations 265(7), 2921\u20132967 (2018). https:\/\/doi.org\/10.1016\/j.jde.2018.04.052","journal-title":"J. Differential Equations"},{"issue":"3","key":"3354_CR23","doi-asserted-by":"publisher","first-page":"1435","DOI":"10.1137\/17M1143927","volume":"56","author":"X Li","year":"2018","unstructured":"Li, X., Ma, Q., Yang, H., Yuan, C.: The numerical invariant measure of stochastic differential equations with Markovian switching. SIAM J. Numer. Anal. 56(3), 1435\u20131455 (2018). https:\/\/doi.org\/10.1137\/17M1143927","journal-title":"SIAM J. Numer. Anal."},{"key":"3354_CR24","doi-asserted-by":"publisher","first-page":"115017","DOI":"10.1016\/j.cam.2022.115017","volume":"424","author":"L Chen","year":"2023","unstructured":"Chen, L., Gan, S.: Strong convergence and stationary distribution of an explicit scheme for the Wright-Fisher model. J. Comput. Appl. Math. 424, 115017\u201322 (2023). https:\/\/doi.org\/10.1016\/j.cam.2022.115017","journal-title":"J. Comput. Appl. Math."},{"issue":"4","key":"3354_CR25","doi-asserted-by":"publisher","first-page":"1531","DOI":"10.1007\/s11075-019-00735-5","volume":"83","author":"Y Jiang","year":"2020","unstructured":"Jiang, Y., Weng, L., Liu, W.: Stationary distribution of the stochastic theta method for nonlinear stochastic differential equations. Numer. Algorithms 83(4), 1531\u20131553 (2020). https:\/\/doi.org\/10.1007\/s11075-019-00735-5","journal-title":"Numer. Algorithms"},{"key":"3354_CR26","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.cam.2014.08.019","volume":"276","author":"W Liu","year":"2015","unstructured":"Liu, W., Mao, X.: Numerical stationary distribution and its convergence for nonlinear stochastic differential equations. J. Comput. Appl. Math. 276, 16\u201329 (2015). https:\/\/doi.org\/10.1016\/j.cam.2014.08.019","journal-title":"J. Comput. Appl. Math."},{"key":"3354_CR27","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1016\/j.apnum.2022.09.017","volume":"184","author":"W Liu","year":"2023","unstructured":"Liu, W., Mao, X., Wu, Y.: The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients. Appl. Numer. Math. 184, 137\u2013150 (2023). https:\/\/doi.org\/10.1016\/j.apnum.2022.09.017","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"3354_CR28","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s10915-025-02902-4","volume":"103","author":"Z Liu","year":"2025","unstructured":"Liu, Z., Liu, Z.: Numerical unique ergodicity of monotone SDEs driven by nondegenerate multiplicative noise. J. Sci. Comput. 103(3), 87\u201321 (2025). https:\/\/doi.org\/10.1007\/s10915-025-02902-4","journal-title":"J. Sci. Comput."},{"key":"3354_CR29","doi-asserted-by":"publisher","first-page":"101842","DOI":"10.1016\/j.jco.2024.101842","volume":"83","author":"C Pang","year":"2024","unstructured":"Pang, C., Wang, X., Wu, Y.: Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients. J. Complexity 83, 101842\u201345 (2024). https:\/\/doi.org\/10.1016\/j.jco.2024.101842","journal-title":"J. Complexity"},{"issue":"5","key":"3354_CR30","doi-asserted-by":"publisher","first-page":"1213","DOI":"10.1016\/j.cam.2010.08.006","volume":"235","author":"X Mao","year":"2011","unstructured":"Mao, X., Shen, Y., Gray, A.: Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations. J. Comput. Appl. Math. 235(5), 1213\u20131226 (2011). https:\/\/doi.org\/10.1016\/j.cam.2010.08.006","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"3354_CR31","doi-asserted-by":"publisher","first-page":"759","DOI":"10.1007\/s10543-019-00793-0","volume":"60","author":"X Wang","year":"2020","unstructured":"Wang, X., Wu, J., Dong, B.: Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition. BIT 60(3), 759\u2013790 (2020). https:\/\/doi.org\/10.1007\/s10543-019-00793-0","journal-title":"BIT"},{"key":"3354_CR32","doi-asserted-by":"publisher","unstructured":"Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood Publishing Limited, Chichester (2008). https:\/\/doi.org\/10.1533\/9780857099402","DOI":"10.1533\/9780857099402"},{"issue":"3","key":"3354_CR33","doi-asserted-by":"publisher","first-page":"1681","DOI":"10.1137\/24M1677113","volume":"47","author":"MV Tretyakov","year":"2025","unstructured":"Tretyakov, M.V.: Sampling from mixture distributions based on regime-switching diffusions. SIAM J. Sci. Comput. 47(3), 1681\u20131701 (2025). https:\/\/doi.org\/10.1137\/24M1677113","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03354-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03354-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03354-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T09:37:42Z","timestamp":1783676262000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03354-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,10]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["3354"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03354-0","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,10]]},"assertion":[{"value":"3 November 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 April 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 June 2026","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 there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"37"}}