Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-28T20:33:36.513Z Has data issue: false hasContentIssue false

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’

Published online by Cambridge University Press:  01 September 2013

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

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Corrigenda
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, 99127.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) 7183.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, 298321.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, 22822293.CrossRefGoogle Scholar