A note on almost sure exponential stability of <i>θ</i> -Euler-Maruyama approximation for neutral stochastic differential equations with time-dependent delay when θ ∈ (12{1 \over 2}, 1)

Abstract This paper is motivated by the paper [2]. The main aim of this paper is to extend the stability result from [16], related to the θ -Euler- Maruyama method ( θ ∈ ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:math> {1 \over 2} , 1)) for a class of neutral stochastic differential equations with time-dependent delay. The theta method is defined such that, in general case, it is implicit in both drift coefficient and neutral term. Sufficient conditions of the a.s. exponential stability of the θ -Euler-Maruyama method, including the linear growth condition on the drift coefficient of the equation, are revealed. The stability result is established for larger class of neutral terms than that considered in the second cited paper. An example is provided to support the main results of the paper.