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