Stability equivalence for stochastic differential equations, stochastic differential delay equations and their corresponding Euler-Maruyama methods in $G$-framework

Abstract

In this paper, we investigate the stability equivalence problem for stochastic differential delay equations, the auxiliary stochastic differential equations and their corresponding Euler-Maruyama (EM) methods under $G$-framework. More precisely, for $p\geq 2$, we prove the equivalence of practical exponential stability in $p$-th moment sense among stochastic differential delay equations driven by $G$-Brownian motion ($G$-SDDEs), the auxiliary stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs), and their corresponding Euler-Maruyama methods, provided the delay or the step size is small enough. Thus, we can carry out careful simulations to examine the practical exponential stability of the underlying $G$-SDDE or $G$-SDE under some reasonable assumptions.