{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:30:43Z","timestamp":1760232643893,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T00:00:00Z","timestamp":1668470400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We introduce two approaches by modifying split-step exponential schemes to study stochastic differential equations. Under the Lipschitz condition and linear-growth bounds, it is shown that our explicit schemes converge to the solution of the corresponding stochastic differential equations with the order 1.0 in the mean-square sense. The mean-square stability of our methods is investigated through some linear stochastic test systems. Additionally, asymptotic mean-square stability is analyzed for the two-dimensional system with symmetric and asymmetric coefficients and driven by two commutative noise terms. In particular, we prove that our methods are mean-square stable for any step-size. Finally, some numerical experiments are carried out to confirm the theoretical results.<\/jats:p>","DOI":"10.3390\/sym14112413","type":"journal-article","created":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T02:31:15Z","timestamp":1668479475000},"page":"2413","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Convergence and Stability of a Split-Step Exponential Scheme Based on the Milstein Methods"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2504-4048","authenticated-orcid":false,"given":"Leila","family":"Torkzadeh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan P.O. Box 35195-363, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9374-1091","authenticated-orcid":false,"given":"Hassan","family":"Ranjbar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan P.O. Box 35195-363, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6233-8299","authenticated-orcid":false,"given":"Sanda","family":"Micula","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Babe\u015f-Bolyai University, 400084 Cluj-Napoca, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7922-5848","authenticated-orcid":false,"given":"Kazem","family":"Nouri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan P.O. Box 35195-363, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ali, I., and Khan, S.U. (2022). Threshold of stochastic SIRS epidemic model from infectious to susceptible class with saturated incidence rate using spectral method. Symmetry, 14.","DOI":"10.3390\/sym14091838"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Eissa, M.A., and Elsayed, M. (2022). Improve stock price model-based stochastic pantograph differential equation. Symmetry, 14.","DOI":"10.3390\/sym14071358"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"George, R., Mitrovic, Z.D., Turab, A., Savic, A., and Ali, W. (2022). On a unique solution of a class of stochastic predator-prey models with two-choice behavior of predator animals. Symmetry, 11.","DOI":"10.3390\/sym14050846"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kloeden, P.E., and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer.","DOI":"10.1007\/978-3-662-12616-5"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Lee, Y., Kim, Y., and Lee, J. (2020). Pricing various types of power options under stochastic volatility. Symmetry, 12.","DOI":"10.3390\/sym12111911"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Geiser, J. (2020). Numerical Picard iteration methods for simulation of non-Lipschitz stochastic differential equations. Symmetry, 3.","DOI":"10.3390\/sym12030383"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s11075-012-9534-5","article-title":"A class of split-step balanced methods for stiff stochastic differential equations","volume":"61","author":"Haghighi","year":"2012","journal-title":"Numer. Algorithms"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1137\/S0036142901389530","article-title":"Strong convergence of Euler-type methods for nonlinear stochastic differential equations","volume":"40","author":"Higham","year":"2002","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Milstein, G.N., and Tretyakov, M.V. (2004). Stochastic Numerics for Mathematical Physics, Springer.","DOI":"10.1007\/978-3-662-10063-9"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1007\/s00009-018-1187-8","article-title":"Improved Euler-Maruyama method for numerical solution of the It\u00f4 stochastic differential systems by composite previous-current-step idea","volume":"15","author":"Nouri","year":"2018","journal-title":"Mediterr. J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.cnsns.2018.08.013","article-title":"Modified stochastic theta methods by ODEs solvers for stochastic differential equations","volume":"68","author":"Nouri","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1198","DOI":"10.1016\/j.apnum.2008.06.001","article-title":"Split-step backward balanced Milstein methods for stiff stochastic systems","volume":"59","author":"Wang","year":"2009","journal-title":"Appl. Numer. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Zhang, E., and Li, L. (2022). The improved stability analysis of numerical method for stochastic delay differential equations. Mathematics, 10.","DOI":"10.3390\/math10183366"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Hairer, E., and Wanner, G. (1991). Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer.","DOI":"10.1007\/978-3-662-09947-6"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1030","DOI":"10.1080\/00207160.2018.1480761","article-title":"Split-step double balanced approximation methods for stiff stochastic differential equations","volume":"96","author":"Haghighi","year":"2019","journal-title":"Int. J. Comput. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1080\/00207160.2019.1589459","article-title":"Study on split-step Rosenbrock type method for stiff stochastic differential systems","volume":"97","author":"Nouri","year":"2020","journal-title":"Int. J. Comput. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"990","DOI":"10.1137\/0910058","article-title":"Efficient split linear multistep methods for stiff ordinary differential equations","volume":"10","author":"Voss","year":"1989","journal-title":"SIAM J. Sci. Statist. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apnum.2014.10.005","article-title":"Split-step Milstein methods for multi-channel stiff stochastic differential systems","volume":"89","author":"Reshniak","year":"2015","journal-title":"Appl. Numer. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1080\/00207160.2014.915963","article-title":"Split-step Adams-Moulton Milstein methods for systems of stiff stochastic differential equations","volume":"92","author":"Voss","year":"2015","journal-title":"Int. J. Comput. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s40096-021-00392-7","article-title":"Improving split-step forward methods by ODE solver for stiff stochastic differential equations","volume":"16","author":"Nouri","year":"2022","journal-title":"Math. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5521","DOI":"10.1016\/j.aej.2021.04.040","article-title":"Investigation on Ginzburg-Landau equation via a tested approach to benchmark stochastic Davis-Skodje system","volume":"60","author":"Nouri","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"758","DOI":"10.1140\/epjp\/s13360-020-00765-2","article-title":"The explicit approximation approach to solve stiff chemical Langevin equations","volume":"135","author":"Nouri","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"7498865","DOI":"10.1155\/2022\/7498865","article-title":"Improvement of split-step forward Milstein schemes for SODEs arising in mathematical physics","volume":"2022","author":"Ranjbar","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2254","DOI":"10.1137\/S0036142992228409","article-title":"Stability analysis of numerical schemes for stochastic differential equations","volume":"33","author":"Saito","year":"1996","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_25","first-page":"551","article-title":"Mean-square stability of numerical schemes for stochastic differential systems","volume":"30","author":"Saito","year":"2002","journal-title":"Vietnam J. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1110","DOI":"10.1016\/j.matcom.2010.09.015","article-title":"A comparative linear mean-square stability analysis of Maruyama and Milstein-type methods","volume":"81","author":"Buckwar","year":"2011","journal-title":"Math. Comput. Simul."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1016\/j.apnum.2012.03.002","article-title":"A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems","volume":"62","author":"Buckwar","year":"2012","journal-title":"Appl. Numer. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1310","DOI":"10.1016\/j.camwa.2010.06.011","article-title":"Convergence and stability of the split-step \u03b8-method for stochastic differential equations","volume":"60","author":"Ding","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1023\/A:1022355410570","article-title":"A-stability and stochastic mean-square stability","volume":"40","author":"Higham","year":"2000","journal-title":"BIT Numer. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2245","DOI":"10.1002\/mma.2972","article-title":"The improved split-step \u03b8 methods for stochastic differential equation","volume":"37","author":"Guo","year":"2014","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Khasminskii, R. (2012). Stochastic Stability of Differential Equations, Springer.","DOI":"10.1007\/978-3-642-23280-0"},{"key":"ref_32","unstructured":"Mao, X. (1997). Stochastic Differential Equations and Their Applications, Horwood Publishing."},{"key":"ref_33","unstructured":"Arnold, L. (1974). Stochastic Differential Equations: Theory and Applications, John Wiley & Sons."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2413\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:18:18Z","timestamp":1760145498000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2413"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,15]]},"references-count":33,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112413"],"URL":"https:\/\/doi.org\/10.3390\/sym14112413","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,11,15]]}}}