Stability equivalence between the neutral delayed stochastic differential equations and the Euler-Maruyama numerical scheme

In this paper, the equivalence theorem for the mean square exponential stability between the neutral delayed stochastic differential equations (NDSDEs) and the Euler-Maruyama numerical scheme is investigated via the continuous time Euler-Maruyama solutions. Firstly, with some preliminaries on basic notations and assumptions, we establish the approximation degree of the numerical scheme to the underlying NDSDE under the global Lipschitz condition for the dynamics and contractive mapping condition for the neutral operator of the equation, which guarantee the existence and uniqueness of the global solution. Then we show that the underlying NDSDE is exponentially stable in mean square if and only if, for some sufficiently small stepsize, the Euler-Maruyama numerical scheme is exponentially stable in mean square. With such a theoretical result, the mean square exponential stability of NDSDEs can be affirmed just by the simulation approach in practice. Finally, a constructive example is proposed to verify the theoretical result by simulation. Relatively, some analysis around the present topic will be given by remarks and some challenging problems for further works will be proposed in the conclusion section.