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.

Automated summary

In this paper, an equivalence relationship of exponential stability is established among stochastic differential equations (SDEs), stochastic differential delay equations (SDDEs), and their corresponding Euler-Maruyama (EM) methods. It is shown that the exponential stability of these four stochastic processes can be deduced from each other if the delay or the step size is sufficiently small. Using this relationship, stability equivalence between SDEs (or SDDEs) and their numerical methods, as well as between delay differential (or difference) equations and the corresponding delay-free equations, can be obtained. This allows for careful numerical calculations to examine equation stability. Additionally, the stability problem for one equation can even be transformed into the stability problem for another if they are close in some sense. This idea provides flexibility in considering equation stability. Finally, an example is provided to illustrate the analytical outcomes.