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Corrigendum: On the use of a discrete form of the Itô formula in the article ‘Almost sure asymptotic stability analysis of the $\theta $-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations’

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

Published online by Cambridge University Press:  01 September 2013

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.jmalexandra.rodkina@uwimona.edu.jm
Alexandra Rodkina
Affiliation:
Department of Mathematics,University of the West Indies,Mona, Kingston 7, Jamaica email conall.kelly@uwimona.edu.jmalexandra.rodkina@uwimona.edu.jm

Abstract

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In the original article [LMS J. Comput. Math. 15 (2012) 71–83], the authors use a discrete form of the Itô formula, developed by Appleby, Berkolaiko and Rodkina [Stochastics 81 (2009) no. 2, 99–127], to show that the almost sure asymptotic stability of a particular two-dimensional test system is preserved when the discretisation step size is small. In this Corrigendum, we identify an implicit assumption in the original proof of the discrete Itô formula that, left unaddressed, would preclude its application to the test system of interest. We resolve this problem by reproving the relevant part of the discrete Itô formula in such a way that confirms its applicability to our test equation. Thus, we reaffirm the main results and conclusions of the original article.

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
Corrigenda
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 366 - 372
Copyright
© The Author(s) 2013 

References

Appleby, J. A. D., Berkolaiko, G. and Rodkina, A., ‘Non-exponential stability and decay rates in nonlinear stochastic difference equation with unbounded noises’, Stochastics 81 (2009) no. 2, 99–127.CrossRefGoogle Scholar
Berkolaiko, G., Buckwar, E., Kelly, C. and Rodkina, A., ‘Almost sure asymptotic stability analysis of the $\theta $ -Maruyama method applied to a test system with stabilising and destabilising perturbations’, LMS J. Comput. Math. 15 (2012) 71–83.Google Scholar
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
Buckwar, E. and Kelly, C., ‘Non-normal drift structures and linear stability analysis of numerical methods for systems of stochastic differential equations’, Comput. Math. Appl. 64 (2012) no. 7, 2282–2293.CrossRefGoogle Scholar