Almost sure stability with general decay rate of exact and numerical solutions for stochastic pantograph differential equations
The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is insufficient for SPDEs. In this paper, the sufficient conditions of th...
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| Published in | Numerical algorithms Vol. 80; no. 4; pp. 1391 - 1411 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.04.2019
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1017-1398 1572-9265 |
| DOI | 10.1007/s11075-018-0531-1 |
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| Abstract | The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is insufficient for SPDEs. In this paper, the sufficient conditions of the almost sure stability with general decay rate of the exact and numerical solutions for SPDEs have been considered. The stability of two different theta numerical methods, namely the split–step theta method and the stochastic linear theta method, have been discussed respectively. From the conditions, we established for the two theta approximations to reproduce the stability of the exact solution, and we see that the conditions for
𝜃
∈
0
,
1
2
are stronger than those for
𝜃
∈
1
2
,
1
. The bound of the Lyapunov exponent of the split-step theta method is little bigger than that of the stochastic linear theta method for sufficiently small step size. To illustrate the theoretical results, we give two examples to examine the almost sure exponential stability. |
|---|---|
| AbstractList | The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is insufficient for SPDEs. In this paper, the sufficient conditions of the almost sure stability with general decay rate of the exact and numerical solutions for SPDEs have been considered. The stability of two different theta numerical methods, namely the split–step theta method and the stochastic linear theta method, have been discussed respectively. From the conditions, we established for the two theta approximations to reproduce the stability of the exact solution, and we see that the conditions for ðoef∈0,12 are stronger than those for ðoef∈12,1. The bound of the Lyapunov exponent of the split-step theta method is little bigger than that of the stochastic linear theta method for sufficiently small step size. To illustrate the theoretical results, we give two examples to examine the almost sure exponential stability. The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is insufficient for SPDEs. In this paper, the sufficient conditions of the almost sure stability with general decay rate of the exact and numerical solutions for SPDEs have been considered. The stability of two different theta numerical methods, namely the split–step theta method and the stochastic linear theta method, have been discussed respectively. From the conditions, we established for the two theta approximations to reproduce the stability of the exact solution, and we see that the conditions for 𝜃 ∈ 0 , 1 2 are stronger than those for 𝜃 ∈ 1 2 , 1 . The bound of the Lyapunov exponent of the split-step theta method is little bigger than that of the stochastic linear theta method for sufficiently small step size. To illustrate the theoretical results, we give two examples to examine the almost sure exponential stability. |
| Author | Guo, Ping Li, Chong–Jun |
| Author_xml | – sequence: 1 givenname: Ping surname: Guo fullname: Guo, Ping organization: School of Mathematical Sciences, Dalian University of Technology – sequence: 2 givenname: Chong–Jun orcidid: 0000-0001-9501-4314 surname: Li fullname: Li, Chong–Jun email: chongjun@dlut.edu.cn organization: School of Mathematical Sciences, Dalian University of Technology |
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| Cites_doi | 10.1016/j.cam.2013.03.038 10.1080/07362990500118637 10.1016/j.cam.2011.09.045 10.1016/j.amc.2015.04.022 10.1016/j.cam.2012.03.005 10.1007/s11075-016-0162-3 10.1166/jama.2016.1099 10.1515/ROSE.2011.007 10.1080/10236198.2014.892934 10.1016/j.jmaa.2009.02.011 10.1016/j.cam.2015.03.016 10.1016/j.jmaa.2006.02.063 10.1007/s11075-012-9534-5 10.1016/j.jmaa.2005.03.056 10.1002/rnc.1726 10.1016/j.jfranklin.2014.02.004 10.1016/j.amc.2014.03.132 10.14232/ejqtde.2016.8.2 10.1007/s00211-010-0294-7 10.1137/060658138 10.1137/S0036142901389530 10.1016/j.amc.2010.12.023 10.1080/00207160.2017.1299862 10.1016/j.jmaa.2017.10.002 10.1016/j.sysconle.2012.11.009 |
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| Keywords | Stochastic linear theta method 60H10 General decay rate 65C30 65L20 Stochastic pantograph differential equation Almost sure stability Split-step theta method |
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Appl.20093552811820252175510.1016/j.jmaa.2009.02.0111166.60040 WuFHuSRazumikhin-type theorems on general decay stability and robustness for stochastic functional differential equationsInt. J. Robust Nonlinear Control2012227763777290891510.1002/rnc.17261276.93081 HighamDJMaoXYuanCAlmost sure and moment exponential stability in the numerical simulation of stochastic differential equationsSIAM J. Numer. Anal.2007452592609230028910.1137/0606581381144.65005 MiloševićMExistence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler-Maruyama approximationAppl. Math. Comput.201423767268532011641334.60117 Song, M., Lu, Y., Liu, M.: Stability of analytical solutions and convergence of numerical methods for non-linear stochastic pantograph differential equations. arXiv:1502.00061 (2015) MaoXNumerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditionsAppl. Math. Comput.2011217125512552427701691215.65015 HighamDJMaoXStuartAMStrong convergence of Euler-type methods for nonlinear stochastic differential equationsSIAM J. Numer. Anal.200240310411063194940410.1137/S00361429013895301026.65003 MaoXRassiasMJKhasminskii-type theorems for stochastic differential delay equationsStoch. Anal. Appl.200523510451069215889110.1080/073629905001186371082.60055 HuLMaoXShenYStability and boundedness of nonlinear hybrid stochastic differential delay equationsSyst. Control Lett.2013622178187300859410.1016/j.sysconle.2012.11.0091259.93127 ZongXWuFHuangCExponential mean square stability of the theta approximations for neutral stochastic differential delay equationsJ. Comput. Appl. Math.2015286172185333623510.1016/j.cam.2015.03.0161320.34111 WuFMaoXSzpruchLAlmost sure exponential stability of numerical solutions for stochastic delay differential equationsNumer. Math.20101154681697265815910.1007/s00211-010-0294-71193.65009 GuoPLiCAlmost sure exponential stability of numerical solutions for the stochastic pantograph differential equationsJ. Math. Anal. Appl.2018460411424373991310.1016/j.jmaa.2017.10.0021382.65018 HaghighiAHosseiniSMA class of split-step balanced methods for stiff stochastic differential equationsNumerical Algorithms2012611141162295940910.1007/s11075-012-9534-506085765 MaoXStochastic differential equations and applications2007ChichesterHorvood1138.60005 PavlovićGJankovićSRazumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delayJ. Comput. Appl. Math.2012236716791690286350510.1016/j.cam.2011.09.0451247.34128 XiaoYMahmoudATianBConvergence and stability of split-step theta methods with variable step-size for stochastic pantograph differential equationsInt. J. Comput. Math.2017955939960377214510.1080/00207160.2017.1299862 Appleby, J.A.D., Buckwar, E.: Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation. Electronic Journal of Qualitative Theory of Differential Equations. (2), 1–32 (2016) ZhangHXiaoYGuoFConvergence and stability of a numerical method for nonlinear stochastic pantograph equationsJ. Frankl. Inst.2014351630893103320102110.1016/j.jfranklin.2014.02.0041290.93202 XiaoYSongMLiuMConvergence and stability of the semi-implicit euler method with variable stepsize for a linear stochastic pantograph differential equationInt. J. Numer. Anal. Model2011821422527404891214.65003 SchurzHAlmost sure asymptotic stability and convergence of stochastic theta methods applied to systems of linear SDEs in $R^{d}$ RdRandom Operators and Stochastic Equations2011192111129280588110.1515/ROSE.2011.0071268.65009 HuangCExponential mean square stability of numerical methods for systems of stochastic differential equationsJ. Comput. Appl. Math.20122361640164026292625910.1016/j.cam.2012.03.0051251.65003 FanZLiuMCaoWExistence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equationsJ. Math. Anal. Appl.2007325211421159227007510.1016/j.jmaa.2006.02.0631107.60030 MaoXLamJXuSRazumikhin method and exponential stability of hybrid stochastic delay interval systemsJ. Math. Anal. Appl.200631414566218353610.1016/j.jmaa.2005.03.0561127.60072 ZongXWuFHuangCPreserving exponential mean square stability and decay rates in two classes of theta approximations of stochastic differential equationsJ. Differ. Equ. Appl.201420710911111321033210.1080/10236198.2014.8929341291.60143 LiuWMaoXAlmost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equationsNumerical Algorithms2017742573592359966510.1007/s11075-016-0162-31371.65010 H Zhang (531_CR28) 2014; 351 X Zong (531_CR29) 2014; 20 DJ Higham (531_CR6) 2002; 40 X Mao (531_CR16) 2011; 217 F Wu (531_CR24) 2010; 115 X Mao (531_CR15) 2007 H Schurz (531_CR21) 2011; 19 L Hu (531_CR8) 2013; 62 W Liu (531_CR13) 2017; 74 A Mahmoud (531_CR14) 2016; 5 Y Xiao (531_CR25) 2017; 95 Y Xiao (531_CR26) 2011; 8 F Wu (531_CR23) 2012; 22 X Zong (531_CR30) 2015; 286 Z Fan (531_CR2) 2007; 325 M Milošević (531_CR19) 2014; 237 X Mao (531_CR17) 2006; 314 531_CR1 R Khasminskii (531_CR12) 2011 531_CR22 P Guo (531_CR4) 2018; 460 C Huang (531_CR9) 2012; 236 S You (531_CR27) 2015; 263 G Pavlović (531_CR20) 2012; 236 TC Gard (531_CR3) 1988 A Haghighi (531_CR5) 2012; 61 C Huang (531_CR10) 2014; 259 S Janković (531_CR11) 2009; 355 DJ Higham (531_CR7) 2007; 45 X Mao (531_CR18) 2005; 23 |
| References_xml | – reference: GuoPLiCAlmost sure exponential stability of numerical solutions for the stochastic pantograph differential equationsJ. Math. Anal. Appl.2018460411424373991310.1016/j.jmaa.2017.10.0021382.65018 – reference: ZongXWuFHuangCExponential mean square stability of the theta approximations for neutral stochastic differential delay equationsJ. Comput. Appl. Math.2015286172185333623510.1016/j.cam.2015.03.0161320.34111 – reference: KhasminskiiRStochastic stability of differential equations2011BerlinSpringer Science & Business Media – reference: FanZLiuMCaoWExistence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equationsJ. Math. Anal. Appl.2007325211421159227007510.1016/j.jmaa.2006.02.0631107.60030 – reference: MaoXRassiasMJKhasminskii-type theorems for stochastic differential delay equationsStoch. Anal. Appl.200523510451069215889110.1080/073629905001186371082.60055 – reference: MaoXLamJXuSRazumikhin method and exponential stability of hybrid stochastic delay interval systemsJ. Math. Anal. Appl.200631414566218353610.1016/j.jmaa.2005.03.0561127.60072 – reference: LiuWMaoXAlmost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equationsNumerical Algorithms2017742573592359966510.1007/s11075-016-0162-31371.65010 – reference: HuLMaoXShenYStability and boundedness of nonlinear hybrid stochastic differential delay equationsSyst. Control Lett.2013622178187300859410.1016/j.sysconle.2012.11.0091259.93127 – reference: XiaoYMahmoudATianBConvergence and stability of split-step theta methods with variable step-size for stochastic pantograph differential equationsInt. J. Comput. Math.2017955939960377214510.1080/00207160.2017.1299862 – reference: HaghighiAHosseiniSMA class of split-step balanced methods for stiff stochastic differential equationsNumerical Algorithms2012611141162295940910.1007/s11075-012-9534-506085765 – reference: YouSMaoWMaoXAnalysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficientsAppl. Math. Comput.20152637383334852606921648 – reference: Appleby, J.A.D., Buckwar, E.: Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation. Electronic Journal of Qualitative Theory of Differential Equations. (2), 1–32 (2016) – reference: PavlovićGJankovićSRazumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delayJ. Comput. Appl. Math.2012236716791690286350510.1016/j.cam.2011.09.0451247.34128 – reference: MaoXStochastic differential equations and applications2007ChichesterHorvood1138.60005 – reference: MaoXNumerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditionsAppl. Math. Comput.2011217125512552427701691215.65015 – reference: ZongXWuFHuangCPreserving exponential mean square stability and decay rates in two classes of theta approximations of stochastic differential equationsJ. Differ. Equ. Appl.201420710911111321033210.1080/10236198.2014.8929341291.60143 – reference: JankovićSRandjelovićJJovanovićMRazumikhin-type exponential stability criteria of neutral stochastic functional differential equationsJ. Math. Anal. Appl.20093552811820252175510.1016/j.jmaa.2009.02.0111166.60040 – reference: WuFHuSRazumikhin-type theorems on general decay stability and robustness for stochastic functional differential equationsInt. J. Robust Nonlinear Control2012227763777290891510.1002/rnc.17261276.93081 – reference: MahmoudAXiaoYTianBConvergence and stability of two classes of theta methods with variable step size for a stochastic pantograph differential equationsJ. Adv. Math. Appl.2016529510610.1166/jama.2016.1099 – reference: ZhangHXiaoYGuoFConvergence and stability of a numerical method for nonlinear stochastic pantograph equationsJ. Frankl. Inst.2014351630893103320102110.1016/j.jfranklin.2014.02.0041290.93202 – reference: HighamDJMaoXYuanCAlmost sure and moment exponential stability in the numerical simulation of stochastic differential equationsSIAM J. Numer. Anal.2007452592609230028910.1137/0606581381144.65005 – reference: HuangCMean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equationsJ. Comput. Appl. Math.20142597786312347210.1016/j.cam.2013.03.0381291.65020 – reference: XiaoYSongMLiuMConvergence and stability of the semi-implicit euler method with variable stepsize for a linear stochastic pantograph differential equationInt. J. Numer. Anal. Model2011821422527404891214.65003 – reference: HuangCExponential mean square stability of numerical methods for systems of stochastic differential equationsJ. Comput. Appl. 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| SubjectTerms | Algebra Algorithms Approximation Computer Science Decay rate Differential equations Exact solutions Liapunov exponents Numeric Computing Numerical Analysis Numerical methods Original Paper Pantographs Polynomials Stability Theory of Computation |
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| Title | Almost sure stability with general decay rate of exact and numerical solutions for stochastic pantograph differential equations |
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