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 CMFD, 2006, Volume 17, Pages 110–128 (Mi cmfd60)

Almost sure polynomial asymptotic stability of stochastic difference equations

J. Applebya, D. Mackeyb, A. Rodkinac

a Dublin City University
b Dublin Institute of Technology
c University of the West Indies

Abstract: In this paper, we establish the almost sure asymptotic stability and decay results for solutions of an autonomous scalar difference equation with a nonhyperbolic equilibrium at the origin, which is perturbed by a random term with a fading state–independent intensity. In particular, we show that if the unbounded noise has tails which fade more quickly than polynomially, then the state–independent perturbation dies away at a sufficiently fast polynomial rate in time, and if the autonomous difference equation has a polynomial nonlinearity at the origin, then the almost sure polynomial rate of decay of solutions can be determined exactly.

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English version:
Journal of Mathematical Sciences, 2008, 149:6, 1629–1647

Bibliographic databases:

UDC: 517.55+517.95

Citation: J. Appleby, D. Mackey, A. Rodkina, “Almost sure polynomial asymptotic stability of stochastic difference equations”, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, CMFD, 17, PFUR, M., 2006, 110–128; Journal of Mathematical Sciences, 149:6 (2008), 1629–1647

Citation in format AMSBIB
\Bibitem{AppMacRod06} \by J.~Appleby, D.~Mackey, A.~Rodkina \paper Almost sure polynomial asymptotic stability of stochastic difference equations \inbook Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14--21, 2005). Part~3 \serial CMFD \yr 2006 \vol 17 \pages 110--128 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd60} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2336462} \transl \jour Journal of Mathematical Sciences \yr 2008 \vol 149 \issue 6 \pages 1629--1647 \crossref{https://doi.org/10.1007/s10958-008-0086-0} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-40549096004} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Appleby J., Berkolaiko G., Rodkina A., “On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations”, J. Difference Equ. Appl., 14:9 (2008), 923–951
2. Kelly C., Rodkina A., “Constrained stability and instability of polynomial difference equations with state-dependent noise”, Discrete Contin. Dyn. Syst. Ser. B, 11:4 (2009), 913–933
3. Appleby J.A.D., Berkolaiko G., Rodkina A., “Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise”, Stochastics, 81:2 (2009), 99–127
4. Appleby J.A.D., Rodkina A., Schurz H., “Non-positivity and oscillations of solutions of nonlinear stochastic difference equations with state-dependent noise”, J. Difference Equ. Appl., 16:7 (2010), 807–830
5. Berkolaiko G., Kelly C., Rodkina A., “Sharp Pathwise Asymptotic Stability Criteria for Planar Systems of Linear Stochastic Difference Equations”, Discret. Contin. Dyn. Syst., 2011, no. S, SI, 163–173
6. Liu W., Foondun M., Mao X., “Mean Square Polynomial Stability of Numerical Solutions to a Class of Stochastic Differential Equations”, Stat. Probab. Lett., 92 (2014), 173–182
7. Rodkina A., Dokuchaev N., “Instability and Stability of Solutions of Systems of Nonlinear Stochastic Difference Equations with Diagonal Noise”, J. Differ. Equ. Appl., 20:5-6, SI (2014), 744–764
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