Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
BIT Numerical Mathematics
Published in: BIT Numerical Mathematics 1/2019

10-09-2018

Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations

Authors: Ping Guo, Chong-Jun Li

Published in: BIT Numerical Mathematics | Issue 1/2019

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we establish Razumikhin-type theorems on \(\alpha \)th moment polynomial stability of exact solution for the stochastic pantograph differential equations, which improves the existing stochastic Razumikhin-type theorems. By using discrete Razumikhin-type technique, we construct conditions for the stability of general numerical scheme of the stochastic pantograph differential equations (SPDEs). The stabilities mainly conclude the global \(\alpha \)th moment asymptotically stability and \(\alpha \)th moment polynomial stability. Using the conditions constructed for the stability of the numerical solutions, we discuss the stability of two special numerical methods, namely the Euler–Maruyama method and the backward Euler–Maruyama method. Finally, an example is given to illustrate the consistence with the theoretical results on \(\alpha \)th moment polynomial stability.
previous article Krylov integrators for Hamiltonian systems
next article A stabilised finite element method for the plate obstacle problem

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
Buckwar, E., Horváth-Bokor, R., Winkler, R.: Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations. BIT Numer. Math. 46(2), 261–282 (2006)MathSciNetCrossRefMATH
2.
Fan, Z., Song, M., Liu, M.: The \(\alpha \)th moment stability for the stochastic pantograph equation. J. Comput. Appl. Math. 233(2), 109–120 (2009)MathSciNetCrossRefMATH
3.
Guo, P., Li, C.: Almost sure exponential stability of numerical solutions for the stochastic pantograph differential equations. J. Math. Anal. Appl. 460, 411–424 (2018)MathSciNetCrossRefMATH
4.
5.
Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)CrossRefMATH
6.
Hu, L., Mao, X., Shen, Y.: Stability and boundedness of nonlinear hybrid stochastic differential delay equations. Syst. Control Lett. 62(2), 178–187 (2013)MathSciNetCrossRefMATH
7.
Janković, S., Randjelović, J., Jovanović, M.: Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations. J. Math. Anal. Appl. 355(2), 811–820 (2009)MathSciNetCrossRefMATH
8.
Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Academic press, London (1986)
9.
10.
Mao, X.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations. Stoch. Process. Their Appl. 65(2), 233–250 (1996)MathSciNetCrossRefMATH
11.
Mao, X.: Razumikhin-type theorems on exponential stability of neutral stochastic differential equations. SIAM J. Math. Anal. 28(2), 389–401 (1997)MathSciNetCrossRefMATH
12.
Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (2007)MATH
13.
Mao, X., Lam, J., Xu, S.: Razumikhin method and exponential stability of hybrid stochastic delay interval systems. J. Math. Anal. Appl. 314(1), 45–66 (2006)MathSciNetCrossRefMATH
14.
Milošević, M.: Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler–Maruyama approximation. Appl. Math. Comput. 237, 672–685 (2014)MathSciNetMATH
15.
Pavlović, G., Janković, S.: Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay. J. Comput. Appl. Math. 236(7), 1679–1690 (2012)MathSciNetCrossRefMATH
16.
Peng, S., Zhang, Y.: Razumikhin-type theorems on \(p\)th moment exponential stability of impulsive stochastic delay differential equations. IEEE Trans. Automat. Contr. 55(8), 1917–1922 (2010)CrossRefMATH
17.
Razumikhin, B.S.: On the stability of systems with a delay. Prikl. Mat. Mekh 20(4), 500–512 (1956)MathSciNet
18.
Shardlow, T., Taylor, P.: On the pathwise approximation of stochastic differential equations. BIT Numer. Math. 56(3), 1101–1129 (2016)MathSciNetCrossRefMATH
19.
Wang, T., Luo, X., Li, W.: Razumikhin-type approach on state feedback of stochastic high-order nonlinear systems with time-varying delay. Int. J. Robust Nonlinear Control (2017)
20.
Wang, X., Gan, S., Wang, D.: A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise. BIT Numer. Math. 52(3), 741–772 (2012)MathSciNetCrossRefMATH
21.
Wu, F., Hu, S.: Razumikhin-type theorems on general decay stability and robustness for stochastic functional differential equations. Int. J. Robust Nonlinear Control 22(7), 763–777 (2012)MathSciNetCrossRefMATH
22.
Wu, F., Hu, S., Mao, X.: Razumikhin-type theorem for neutral stochastic functional differential equations with unbounded delay. Acta Mathematica Scientia 31(4), 1245–1258 (2011)MathSciNetCrossRefMATH
23.
Wu, F., Mao, X., Kloeden, P.E.: Discrete Razumikhin-type technique and stability of the Euler–Maruyama method to stochastic functional differential equations. Discrete Contin. Dyn. Syst. Ser. A 33(2), 885–903 (2013)MathSciNetCrossRefMATH
24.
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)MathSciNetCrossRefMATH
25.
Yang, Z., Zhu, E., Xu, Y., Tan, Y.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations with infinite delay. Acta Applicandae Mathematicae 111(2), 219–231 (2010)MathSciNetCrossRefMATH
26.
Zhang, H., Xiao, Y., Guo, F.: Convergence and stability of a numerical method for nonlinear stochastic pantograph equations. J. Franklin Inst. 351(6), 3089–3103 (2014)MathSciNetCrossRefMATH
27.
Metadata
Title
Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations
Authors
Ping Guo
Chong-Jun Li
Publication date
10-09-2018
Publisher
Springer Netherlands
Published in
BIT Numerical Mathematics / Issue 1/2019
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-018-0723-z

Other articles of this Issue 1/2019

BIT Numerical Mathematics 1/2019 Go to the issue

OriginalPaper

The time-fractional diffusion inverse problem subject to an extra measurement by a local discontinuous Galerkin method

OriginalPaper

Minimum residual Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems

OriginalPaper

Pricing multi-asset option problems: a Chebyshev pseudo-spectral method

OriginalPaper

A stabilised finite element method for the plate obstacle problem

OriginalPaper

Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations

OriginalPaper

On banded M-splitting iteration methods for solving discretized spatial fractional diffusion equations

Premium Partner

    Image Credits