Almost sure exponential stability of the $\theta$-Euler–Maruyama method, when $\theta \in (\frac{1}{2},1)$, for neutral stochastic differential equations with time-dependent delay under nonlinear growth conditions

Abstract

This paper is motivated by Lan and Yuan (J Comput Appl Math 285:230–242, 2015). The main aim of this paper is to establish certain results for the $\theta$-Euler–Maruyama method for a class of neutral stochastic differential equations with time-dependent delay. The method is defined such that, in general case, it is implicit in both drift and neutral term. The drift and neutral term are both parameterized by $\theta$ in a way which guarantees that for $\theta =0$ and $\theta =1$, the method reduces to the Euler–Maruyama method and backward Euler method, respectively, which can be found in the literature. The one-sided Lipschitz conditions in the present-state and delayed arguments of the drift coefficient of this class of equations for any $\theta \in [0,1]$ are employed in order to guarantee the existence and uniqueness of the appropriate $\theta$-Euler–Maruyama approximate solution. The main result of this paper is almost sure exponential stability of the $\theta$-Euler–Maruyama method, for $\theta \in (\frac{1}{2},1),$ under nonlinear growth conditions. Some comments and conclusions are presented for the corresponding deterministic case. An example and numerical simulations are provided to support the main results of the paper.