Equivalence of stability among stochastic differential equations, stochastic differential delay equations, and their corresponding euler-maruyama methods

An equivalence of the exponential stability concerning stochastic differential equations (SDEs), stochastic differential delay equations (SDDEs), and their corresponding Euler-Maruyama (EM) methods, is established. We show that the exponential stability for these four stochastic processes can be deduced from each other, provided that the delay or the step size is small enough. Using this relationship, we can obtain stability equivalence between SDEs (or SDDEs) and their numerical methods and between delay differential (or difference) equations and the corresponding delay-free equations. Thus, we can perform careful numerical calculations to examine the stability of an equation. On the other hand, we can even transform the problem of the stability for one equation into the stability for another, provided that the two are "close" in some sense. This idea can allow us to be more flexible in considering the stability of equations. Finally, we give an example to show the analytical outcomes.