胃-Maruyama methods for nonlinear stochastic differential delay equations

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• 作者：Xiaojie Wang^a ; ^{x.j.wang7@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Siqing Gan^a ; ^{sqgan@csu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Desheng Wang^b ; ^{Desheng@ntu.edu.sg" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlinear stochastic differential delay equations ; Variable lag ; 胃-Maruyama methods ; Strong convergence ; Exponential mean-square stability ; Mean-square stability
• 刊名：Applied Numerical Mathematics
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：98
• 期：Complete
• 页码：38-58
• 全文大小：551 K

文摘

In this paper, mean-square convergence and mean-square stability of 胃  -Maruyama methods are studied for nonlinear stochastic differential delay equations (SDDEs) with variable lag. Under global Lipschitz conditions, the methods are proved to be mean-square convergent with order mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0168927415001166&_mathId=si1.gif&_user=111111111&_pii=S0168927415001166&_rdoc=1&_issn=01689274&md5=649815b33d96d9fe7eae7e3f0de26857">mage" height="19" width="8" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0168927415001166-si1.gif">mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">12math>, and exponential mean-square stability of SDDEs implies that of the methods for sufficiently small step size mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0168927415001166&_mathId=si2.gif&_user=111111111&_pii=S0168927415001166&_rdoc=1&_issn=01689274&md5=2d9d6b59ab2afc91414004f635defa22" title="Click to view the MathML source">h>0mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">h>0math>. Further, the exponential mean-square stability properties of SDDEs and those of numerical methods are investigated under some non-global Lipschitz conditions on the drift term. It is shown in this setting that the 胃  -Maruyama method with mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0168927415001166&_mathId=si272.gif&_user=111111111&_pii=S0168927415001166&_rdoc=1&_issn=01689274&md5=fe8df85490d50de3da1d4c2cf0f07c67" title="Click to view the MathML source">胃=1mathContainer hidden">mathCode"><math altimg="si272.gif" overflow="scroll">胃=1math> can preserve the exponential mean-square stability for any step size. Additionally, the 胃  -Maruyama method with mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0168927415001166&_mathId=si345.gif&_user=111111111&_pii=S0168927415001166&_rdoc=1&_issn=01689274&md5=532c24aefc0f79eac8a3791d999f10ea">mage" height="19" width="57" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0168927415001166-si345.gif">mathContainer hidden">mathCode"><math altimg="si345.gif" overflow="scroll">12≤胃≤1math> is asymptotically mean-square stable for any step size, provided that the underlying system with constant lag is exponentially mean-square stable. Applications of this work to some special problem classes show that the results are deeper or sharper than those in the literature. Finally, numerical experiments are included to demonstrate the obtained theoretical results.