Advanced search
Publication Cover
Journal of Difference Equations and Applications Volume 20, 2014 - Issue 5-6: Tribute to Gerry Ladas
Submit an article Journal homepage
134
Views
6
CrossRef citations to date
0
Altmetric
Articles

Instability and stability of solutions of systems of nonlinear stochastic difference equations with diagonal noise

Alexandra RodkinaDepartment of Mathematics, The University of the West Indies, Kingston 7, JamaicaCorrespondence[email protected]
&
Nikolai DokuchaevDepartment of Mathematics and Statistics, Curtin University, Perth, Australia
Pages 744-764 | Received 16 Mar 2013, Accepted 06 Jun 2013, Published online: 22 Jul 2013

REFERENCES

  • J.A.D.Appleby, G.Berkolaiko, and A.Rodkina, On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations, J. Difference Equ. Appl.14(9) (2008), pp. 923–951.
  • J.A.D.Appleby, G.Berkolaiko, and A.Rodkina, Non-exponential stability and decay rates in nonlinear stochastic difference equation with unbounded noises, Stoch. Int. J. Probab. Stoch. Process.81(2) (2009), pp. 99–127.
  • J.A.D.Appleby, C.Kelly, X.Mao, and A.Rodkina, On the local stability and instability of polynomial difference equations with fading stochastic perturbations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.17 (2010), pp. 401–430.
  • J.A.D.Appleby, D.Mackey, and A.Rodkina, Almost sure polynomial asymptotic stability of stochastic difference equations, J. Math. Sci.149(6) (2008), pp. 1629–1647.
  • J.A.D.Appleby, X.Mao, and A.Rodkina, On stochastic stabilization of difference equations, Discrete Contin. Dyn. Syst. A15(3) (2006), pp. 843–857.
  • G.Berkolaiko, E.Buckwar, C.Kelly, and A.Rodkina, Almost sure asymptotic stability of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations, LMS J. Comput. Math.15 (2012), pp. 71–83.
  • G.Berkolaiko, C.Kelly, and A.Rodkina, Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations, Discrete Contin. Dyn. Syst.Supplements 2011 Issue Special (2011), pp. 163–173.
  • G.Berkolaiko and A.Rodkina, Almost sure convergence of solutions to non-homogeneous stochastic difference equation, Difference Equ. Appl.12(6) (2006), pp. 535–553.
  • H.Furstenberg and H.Kesten, Products of random matrix, Ann. Math. Stat.31(2) (1960), pp. 457–469.
  • D.J.Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal.38(3) (2003), pp. 753–769.
  • D.J.Higham, X.Mao, and A.M.Stuart, Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal.40(3) (2002), pp. 1041–1063.
  • D.J.Higham, X.Mao, and C.Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal.45 (2007), pp. 592–609.
  • I.Karatzas and S.Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 2nd ed., Springer, Berlin, 1998.
  • C. Kelly, P. Palmer, and A. Rodkina, Almost sure instability of the equilibrium solution of a Milstein-type stochastic difference equation, Computers and Mathematics with Applications, 2013.
  • H.Kesten, Random difference equations and renewal theory for the product of random matrices, Acta Math.131 (1973), pp. 207–248.
  • X.Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood Publishing Limited, Chichester, 1997.
  • A.Rodkina and M.Basin, On delay-dependent stability for vector nonlinear stochastic delay-difference equations with Volterra diffusion term, Syst. Control Lett.56(6) (2007), pp. 423–430.
  • A.Rodkina and X.Mao, On boundedness and stability of solutions of nonlinear difference equation with nonmartingale type noise, J. Difference Equ. Appl. (2001), pp. 7529–7550.
  • A.N.Shiryaev, Probability, 2nd ed., Springer, Berlin, 1996.
  • D.Williams, Probability with Martingales, Cambridge Mathematical Text books, Cambridge University Press, Cambridge, 1991.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.