{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:50:51Z","timestamp":1747198251509,"version":"3.40.5"},"reference-count":14,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2022,2,15]],"date-time":"2022-02-15T00:00:00Z","timestamp":1644883200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,3,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of <jats:inline-formula id=\"j_mcma-2022-2102_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi mathvariant=\"script\">\u2112<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_mcma-2022-2102_eq_0210.png\"\/>\n                        <jats:tex-math>{\\mathcal{L}^{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-convergence of the truncated SD method and showed that it can be arbitrarily close to <jats:inline-formula id=\"j_mcma-2022-2102_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mfrac>\n                              <m:mn>1<\/m:mn>\n                              <m:mn>2<\/m:mn>\n                           <\/m:mfrac>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_mcma-2022-2102_eq_0189.png\"\/>\n                        <jats:tex-math>{\\frac{1}{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>; see\n[I. S. Stamatiou and N. Halidias,\nConvergence rates of the semi-discrete method for stochastic differential equations,\nTheory Stoch. Process. 24 2019, 2, 89\u2013100].\nWe show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE.\nMotivated by a numerical example, we also propose a different SD scheme, using the Lamperti transformation\nto the original SDE. Numerical simulations support our theoretical findings.<\/jats:p>","DOI":"10.1515\/mcma-2022-2102","type":"journal-article","created":{"date-parts":[[2022,2,14]],"date-time":"2022-02-14T12:03:57Z","timestamp":1644840237000},"page":"13-25","source":"Crossref","is-referenced-by-count":1,"title":["A note on the asymptotic stability of the semi-discrete method for stochastic differential equations"],"prefix":"10.1515","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8756-8229","authenticated-orcid":false,"given":"Nikolaos","family":"Halidias","sequence":"first","affiliation":[{"name":"Department of Statistics and Actuarial-Financial Mathematics , University of the Aegean , Mytilini , Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8215-9634","authenticated-orcid":false,"given":"Ioannis S.","family":"Stamatiou","sequence":"additional","affiliation":[{"name":"Department of Biomedical Sciences , University of West Attica , Athens , Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2022,2,15]]},"reference":[{"key":"2023040102234871567_j_mcma-2022-2102_ref_001","doi-asserted-by":"crossref","unstructured":"N.  Halidias,\nSemi-discrete approximations for stochastic differential equations and applications,\nInt. J. Comput. Math. 89 (2012), no. 6, 780\u2013794.","DOI":"10.1080\/00207160.2012.658380"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_002","doi-asserted-by":"crossref","unstructured":"N.  Halidias,\nA novel approach to construct numerical methods for stochastic differential equations,\nNumer. Algorithms 66 (2014), no. 1, 79\u201387.","DOI":"10.1007\/s11075-013-9724-9"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_003","doi-asserted-by":"crossref","unstructured":"N.  Halidias,\nConstructing positivity preserving numerical schemes for the two-factor CIR model,\nMonte Carlo Methods Appl. 21 (2015), no. 4, 313\u2013323.","DOI":"10.1515\/mcma-2015-0109"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_004","doi-asserted-by":"crossref","unstructured":"N.  Halidias,\nConstruction of positivity preserving numerical schemes for some multidimensional stochastic differential equations,\nDiscrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 1, 153\u2013160.","DOI":"10.3934\/dcdsb.2015.20.153"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_005","doi-asserted-by":"crossref","unstructured":"N.  Halidias and I. S.  Stamatiou,\nApproximating explicitly the mean-reverting CEV process,\nJ. Probab. Stat. 2015 (2015), Article ID 513137, 20 pages.","DOI":"10.1155\/2015\/513137"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_006","doi-asserted-by":"crossref","unstructured":"N.  Halidias and I. S.  Stamatiou,\nOn the numerical solution of some non-linear stochastic differential equations using the semi-discrete method,\nComput. Methods Appl. Math. 16 (2016), no. 1, 105\u2013132.","DOI":"10.1515\/cmam-2015-0028"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_007","doi-asserted-by":"crossref","unstructured":"L.  Hu, X.  Li and X.  Mao,\nConvergence rate and stability of the truncated Euler\u2013Maruyama method for stochastic differential equations,\nJ. Comput. Appl. Math. 337 (2018), 274\u2013289.","DOI":"10.1016\/j.cam.2018.01.017"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_008","doi-asserted-by":"crossref","unstructured":"M.  Hutzenthaler, A.  Jentzen and P. E.  Kloeden,\nStrong and weak divergence in finite time of Euler\u2019s method for stochastic differential equations with non-globally Lipschitz continuous coefficients,\nProc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2130, 1563\u20131576.","DOI":"10.1098\/rspa.2010.0348"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_009","doi-asserted-by":"crossref","unstructured":"P. E.  Kloeden and E.  Platen,\nNumerical Solution of Stochastic Differential Equations,\nAppl. Math. (New York) 23,\nSpringer, Berlin, 1992.","DOI":"10.1007\/978-3-662-12616-5"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_010","doi-asserted-by":"crossref","unstructured":"R. S.  Liptser and A. N.  Shiryayev,\nTheory of Martingales,\nMath. Appl. 49,\nSpringer, Cham, 1989.","DOI":"10.1007\/978-94-009-2438-3"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_011","doi-asserted-by":"crossref","unstructured":"X.  Mao,\nStochastic Differential Equations and Applications, 2nd ed.,\nHorwood, Chichester, 2008.","DOI":"10.1533\/9780857099402"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_012","doi-asserted-by":"crossref","unstructured":"I. S.  Stamatiou,\nA boundary preserving numerical scheme for the Wright\u2013Fisher model,\nJ. Comput. Appl. Math. 328 (2018), 132\u2013150.","DOI":"10.1016\/j.cam.2017.07.011"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_013","doi-asserted-by":"crossref","unstructured":"I. S.  Stamatiou,\nAn explicit positivity preserving numerical scheme for CIR\/CEV type delay models with jump,\nJ. Comput. Appl. Math. 360 (2019), 78\u201398.","DOI":"10.1016\/j.cam.2019.04.005"},{"key":"2023040102234871567_j_mcma-2022-2102_ref_014","unstructured":"I. S.  Stamatiou and N.  Halidias,\nConvergence rates of the semi-discrete method for stochastic differential equations,\nTheory Stoch. Process. 24 (2019), no. 2, 89\u2013100."}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2022-2102\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2022-2102\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,2]],"date-time":"2023-04-02T02:08:52Z","timestamp":1680401332000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2022-2102\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,15]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,2,15]]},"published-print":{"date-parts":[[2022,3,1]]}},"alternative-id":["10.1515\/mcma-2022-2102"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2022-2102","relation":{},"ISSN":["0929-9629","1569-3961"],"issn-type":[{"type":"print","value":"0929-9629"},{"type":"electronic","value":"1569-3961"}],"subject":[],"published":{"date-parts":[[2022,2,15]]}}}