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

TL;DR

This paper establishes the stability equivalence between stochastic differential delay equations, their auxiliary equations, and Euler-Maruyama methods within the $G$-framework, enabling reliable simulations of stability properties.

Contribution

It proves the practical exponential stability equivalence among $G$-SDDEs, $G$-SDEs, and their Euler-Maruyama methods under small delay or step size, within the $G$-framework.

Findings

Proves stability equivalence in $p$-th moment sense.

Enables simulation-based stability analysis.

Applicable under small delay or step size.

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.