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

Abstract

In the original article [LMS J. Comput. Math. 15 (2012) 71–83], the authors use a discrete form of the Ito 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 Ito 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 Ito 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.