Document Zbl 1460.65007

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Difference Equ. Appl.</a> <a href="/?q=in%3A459253" title="Articles in this Issue">26, No. 11-12, 1538-1563 (2020)</a>. </div> <div class="abstract">Summary: This paper applies an explicit numerical method, i.e. the truncated Euler-Maruyama (EM) scheme, to investigate numerical approximations for nonlinear and non-autonomous stochastic differential equations (SDEs). The explicit method reproduces asymptotic stability of the equations. Under a weak condition and with locally Lipschitz continuous coefficients, strong convergence of the truncated EM scheme for the SDEs is proved. Combing the Hölder continuous condition for temporal variables, we obtain the convergence rate that is related to the Hölder continuity. Moreover, employing the discrete nonnegative semimartingale convergence theorem, we reformulate the explicit EM scheme to approximate asymptotic stability in an infinite horizon. Numerical simulations are provided to illustrate the validity of our theoretical results.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7314947">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65C30" title="MSC2020">65C30</a> </td> <td class="space"> Numerical solutions to stochastic differential and integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60H10" title="MSC2020">60H10</a> </td> <td class="space"> Stochastic ordinary differential equations (aspects of stochastic analysis) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60H35" title="MSC2020">60H35</a> </td> <td class="space"> Computational methods for stochastic equations (aspects of stochastic analysis) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Alocally+Lipschitz+condition">locally Lipschitz condition</a>; <a href="/?q=ut%3Astrong+convergence">strong convergence</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Anon-autonomous+stochastic+differential+equations">non-autonomous stochastic differential equations</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1460.65007" data-ciurl="/ci/07314947" data-biburl="/bibtex/07314947.bib" data-amsurl="/amsrefs/07314947.bib" data-xmlurl="/xml/07314947.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07314947.pdf" title="Zbl 1460.65007 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1080/10236198.2020.1857748" aria-label="DOI for “Convergence and asymptotic stability of an explicit numerical method for non-autonomous stochastic differential equations”" title="10.1080/10236198.2020.1857748">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Carletti, M., Numerical solution of stochastic differential problems in the biosciences, J. Comput. Appl. Math., 185, 2, 422-440 (2006) · <a href="/1077.65007" class="nowrap">Zbl 1077.65007</a></td> </tr><tr> <td>[2]</td> <td class="space">Ding, X.; Ma, Q.; Zhang, L., Convergence and stability of the split-step-method for stochastic differential equations, Comput. Math. Appl., 60, 5, 1310-1321 (2010) · <a href="/1201.65010" class="nowrap">Zbl 1201.65010</a></td> </tr><tr> <td>[3]</td> <td class="space">Higham, D.; Mao, X.; Stuart, A., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40, 3, 1041-1063 (2002) · <a href="/1026.65003" class="nowrap">Zbl 1026.65003</a></td> </tr><tr> <td>[4]</td> <td class="space">Hutzenthaler, M.; Kloeden, J., Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. Math. Phys. Eng. Sci., 467, 2130, 1563-1576 (2011) · <a href="/1228.65014" class="nowrap">Zbl 1228.65014</a></td> </tr><tr> <td>[5]</td> <td class="space">Hutzenthaler, M.; Jentzen, A.; Kloeden, P., Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probabt., 22, 4, 1611-1641 (2012) · <a href="/1256.65003" class="nowrap">Zbl 1256.65003</a></td> </tr><tr> <td>[6]</td> <td class="space">Li, X.; Mao, X., The improved LaSalle-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 30, 4, 568-589 (2012) · <a href="/1248.60065" class="nowrap">Zbl 1248.60065</a></td> </tr><tr> <td>[7]</td> <td class="space">Li, X., Mao, X., Yang, H., Strong convergence and asymptotic stability of explicit numerical schemes for nonlinear stochastic differential equations. 2020, Available at https://arxiv.org/pdf/2002.06756.pdf.</td> </tr><tr> <td>[8]</td> <td class="space">Li, X.; Mao, X.; Yin, G., Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: Truncation methods, convergence in pth moment and stability, IMA. J. Numer. Anal., 39, 2, 847-892 (2018) · <a href="/1461.65007" class="nowrap">Zbl 1461.65007</a></td> </tr><tr> <td>[9]</td> <td class="space">Lipster, R.; Shiryayev, A., Theory of Martingales (1989), Kluwer Academic Publishers: Kluwer Academic Publishers, Boston, London · <a href="/0728.60048" class="nowrap">Zbl 0728.60048</a></td> </tr><tr> <td>[10]</td> <td class="space">Liu, W.; Mao, X., Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations, Appl. Math. Comput., 223, 4, 389-400 (2013) · <a href="/1329.65018" class="nowrap">Zbl 1329.65018</a></td> </tr><tr> <td>[11]</td> <td class="space">Liu, W.; Mao, X., Almost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equations, Numer. Algorithms, 74, 2, 573-592 (2016) · <a href="/1371.65010" class="nowrap">Zbl 1371.65010</a></td> </tr><tr> <td>[12]</td> <td class="space">Liu, L.; Shen, Y., The almost sure asymptotic stability and pth moment asymptotic stability of nonlinear stochastic delay differential systems with polynomial growth, Asian J. Control, 14, 3, 859-867 (2012) · <a href="/1303.93149" class="nowrap">Zbl 1303.93149</a></td> </tr><tr> <td>[13]</td> <td class="space">Liu, W.; Mao, X.; Tang, J.; Wu, Y., Truncated Euler-Maruyama method for a class of non-autonomous stochastic differential equations, Appl. Numer. Math, 153, 66-81 (2020) · <a href="/1456.65007" class="nowrap">Zbl 1456.65007</a></td> </tr><tr> <td>[14]</td> <td class="space">Mao, X., Stochastic Differential Equations and Applications (1997), Horwood: Horwood, Chichester · <a href="/0884.60052" class="nowrap">Zbl 0884.60052</a></td> </tr><tr> <td>[15]</td> <td class="space">Mao, X., Some contributions to stochastic asymptotic stability and boundedness via multiple Lyapunov functions, J. Math. Anal. Appl., 260, 2, 325-340 (2001) · <a href="/0983.60055" class="nowrap">Zbl 0983.60055</a></td> </tr><tr> <td>[16]</td> <td class="space">Mao, X., The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290, 370-384 (2015) · <a href="/1330.65016" class="nowrap">Zbl 1330.65016</a></td> </tr><tr> <td>[17]</td> <td class="space">Mao, X.; Rassias, M. J., Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23, 5, 1045-1069 (2005) · <a href="/1082.60055" class="nowrap">Zbl 1082.60055</a></td> </tr><tr> <td>[18]</td> <td class="space">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, 1213-1226 (2011) · <a href="/1227.65012" class="nowrap">Zbl 1227.65012</a></td> </tr><tr> <td>[19]</td> <td class="space">Mo, H., Deng, F. and Ren, H., Almost Sure and Moment Exponential Stability of Euler-Maruyama Method for Non-autonomous Stochastic Differential Equations, in Control Conference, IEEE, 2015.</td> </tr><tr> <td>[20]</td> <td class="space">Platen, E.; Shi, L., On the numerical stability of simulation methods for SDEs under multiplicative noise in finance, Quant. Financ., 13, 2, 183-194 (2013) · <a href="/1280.91193" class="nowrap">Zbl 1280.91193</a></td> </tr><tr> <td>[21]</td> <td class="space">Sabanis, S., A note on tamed Euler approximations, Electron. Commun. Prob., 18, 1-10 (2013) · <a href="/1329.60237" class="nowrap">Zbl 1329.60237</a></td> </tr><tr> <td>[22]</td> <td class="space">Sickenberger, T., Mean-square convergence of stochastic multi-step methods with variable step-size, J. Comput. Appl. Math., 212, 2, 300-319 (2008) · <a href="/1131.60064" class="nowrap">Zbl 1131.60064</a></td> </tr><tr> <td>[23]</td> <td class="space">Situ, R., Theory of stochastic differential equations with jumps and applications, J. Food Sci., 24, 5, 552-563 (2006)</td> </tr><tr> <td>[24]</td> <td class="space">Tong, J.; Ma, X.; Zhang, Z.; Zhu, E., Ergodicity for population dynamics driven by stable processes with Markovian switching, Commun. Stat.-Theory Methods, 48, 10, 2446-2458 (2019) · <a href="/07530032" class="nowrap">Zbl 07530032</a></td> </tr><tr> <td>[25]</td> <td class="space">Wang, X.; Gan, S., The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Differ. Equ. Appl., 19, 3, 466-490 (2013) · <a href="/1262.65008" class="nowrap">Zbl 1262.65008</a></td> </tr><tr> <td>[26]</td> <td class="space">Wu, F.; Mao, X.; Szpruch, L., Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math., 115, 4, 681-697 (2010) · <a href="/1193.65009" class="nowrap">Zbl 1193.65009</a></td> </tr><tr> <td>[27]</td> <td class="space">Yang, H.; Li, X., Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons, J. Differ. Equ., 265, 2921-2967 (2018) · <a href="/1391.65148" class="nowrap">Zbl 1391.65148</a></td> </tr><tr> <td>[28]</td> <td class="space">Yu, Z.; Liu, M., Almost surely asymptotic stability of numerical solutions for neutral stochastic delay differential equations, Discrete Dyn. Nat. Soc., 2011 (2011) · <a href="/1252.65018" class="nowrap">Zbl 1252.65018</a></td> </tr><tr> <td>[29]</td> <td class="space">Yue, C., High-order split-step theta methods for non-autonomous stochastic differential equations with non-globally Lipschitz continuous coefficients, Math. Method Appl. Sci., 39, 9, 2380-2400 (2016) · <a href="/1338.65010" class="nowrap">Zbl 1338.65010</a></td> </tr><tr> <td>[30]</td> <td class="space">Yue, C.; Huang, C.; Jiang, F., Strong convergence of split-step theta methods for non-autonomous stochastic differential equations, Int. J. Comput. Math., 91, 10, 2260-2275 (2014) · <a href="/1309.65011" class="nowrap">Zbl 1309.65011</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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