Almost sure exponential stability of the -Euler-Maruyama method, when (12,1), for neutral stochastic differential equations with time-dependent delay under nonlinear growth conditions

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 -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 in a way which guarantees that for =0 and =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 [0,1] are employed in order to guarantee the existence and uniqueness of the appropriate -Euler-Maruyama approximate solution. The main result of this paper is almost sure exponential stability of the -Euler-Maruyama method, for (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.