Mean-square stability of Euler method for nonlinear neutral stochastic delay differential equations

Stochastic differential equations can always simulate the scientific problem in practical truthfully. They have been widely used in Physics, Chemistry, Cybernetics, Finance, Neural Networks, Bionomics, etc. So far there are not many results on the numerical stability of nonlinear neutral stochastic delay differential equations. The purpose of our work is to show that the Euler method applied to the nonlinear neutral stochastic delay differential equations is mean square stable under the condition which guarantees the stability of the analytical solution. The main aim of this paper is to establish new results on the numerical stability. It is proved that the Euler method is mean-square stable under suitable condition, i.e., assume the some conditions are satisfied, then, the Euler method applied to the nonlinear neutral stochastic delay differential equations with initial data is mean-square stable. Moreover, the theoretical result is also verified by a numerical example.