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Almost sure asymptotic stability analysis of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations

Part of: Stochastic analysis Stochastic systems and control Probabilistic methods, simulation and stochastic differential equations Equations and systems with randomness

Published online by Cambridge University Press:  01 April 2012

Gregory Berkolaiko ,
Evelyn Buckwar ,
Cónall Kelly  and
Alexandra Rodkina
Gregory Berkolaiko
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA (email: gregory.berkolaiko@math.tamu.edu)
Evelyn Buckwar
Affiliation:
Institute for Stochastics, Johannes Kepler University Linz, Altenberger Straße 69, 4040 Linz, Austria (email: evelyn.buckwar@jku.at)
Cónall Kelly
Affiliation:
Department of Mathematics, University of the West Indies, Mona, Kingston 7, Jamaica (email: conall.kelly@uwimona.edu.jm)
Alexandra Rodkina
Affiliation:
Department of Mathematics, University of the West Indies, Mona, Kingston 7, Jamaica (email: alexandra.rodkina@uwimona.edu.jm)

Abstract

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We perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our test equation a two-dimensional system of Itô differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure asymptotic stability of the equilibrium solution. For small values of the constant step-size parameter, we derive close-to-sharp conditions for the almost sure asymptotic stability and instability of the equilibrium solution of the discretisation that match those of the original test system. Our investigation demonstrates the use of a discrete form of the Itô formula in the context of an almost sure linear stability analysis.

MSC classification

Secondary: 60H35: Computational methods for stochastic equations 34F05: Equations and systems with randomness 60H10: Stochastic ordinary differential equations 65C30: Stochastic differential and integral equations 93E15: Stochastic stability
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 71 - 83
Copyright
Copyright © London Mathematical Society 2012

References

[1]Akahori, J., ‘Discrete Itô formulas and their applications to stochastic numerics’, RIMS Kôkyûroku Bessatsu 1462 (2006) 202–210.Google Scholar
[2]Appleby, J. A. D., Berkolaiko, G. and Rodkina, A., ‘Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noises’, Stochastics 81 (2009) no. 2, 99–127.CrossRefGoogle Scholar
[3]Appleby, J. A. D., Mao, X. and Rodkina, A., ‘Stabilisation and destabilisation of nonlinear differential equations by noise’, IEEE Trans. Automat. Control 53 (2008) 683–691.CrossRefGoogle Scholar
[4]Berkolaiko, G. and Rodkina, A., ‘On asymptotic behavior of solutions to linear discrete stochastic equation’, Proceedings of the International Conference 2004 – Dynamical Systems and Applications, Antalya, Turkey, 5–10 July 2004. 614–623.Google Scholar
[5]Buckwar, E. and Kelly, C., ‘Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations’, SIAM J. Numer. Anal. 48 (2010) no. 1, 298–321.CrossRefGoogle Scholar
[6]Buckwar, E. and Sickenberger, Th., ‘A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods’, Math. Comput. Simulation 81 (2011) no. 6, 1110–1127.CrossRefGoogle Scholar
[7]Furstenberg, H. and Kesten, H., ‘Products of random matrices’, Ann. Math. Statist. 31 (1960) no. 2, 457–469.CrossRefGoogle Scholar
[8]Grimmett, G. and Stirzaker, D., Probability and random processes (Oxford University Press, Oxford, 1991).Google Scholar
[9]Gzyl, H., ‘An exposé on discrete Wiener chaos expansions’, Bol. Asoc. Mat. Venez. XIII (2006) no. 1, 3–26.Google Scholar
[10]Hairer, E. and Wanner, G., Solving ordinary differential equations. II: Stiff and differential–algebraic problems (Springer, Berlin, 1996).CrossRefGoogle Scholar
[11]Higham, D. J., ‘Mean-square and asymptotic stability of the stochastic theta method’, SIAM J. Numer. Anal. 38 (2000) no. 3, 753–769.CrossRefGoogle Scholar
[12]Higham, D. J., Mao, X. and Yuan, C., ‘Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations’, SIAM J. Numer. Anal. 45 (2007) 592–609.CrossRefGoogle Scholar
[13]Kannan, D. and Zhang, B., ‘A discrete-time Itô’s formula’, Stoch. Anal. Appl. 20 (2002) no. 5, 1133–1140.CrossRefGoogle Scholar
[14]Kesten, H., ‘Random difference equations and renewal theory for the product of random matrices’, Acta Math. 131 (1973) 207–248.CrossRefGoogle Scholar
[15]Mao, X., Stochastic differential equations and their applications (Horwood, Chichester, 1997).Google Scholar
[16]Shiryaev, A. N., Probability, 2nd edn (Springer, Berlin, 1996).CrossRefGoogle Scholar