Quasi-Sure Exponential Stability of Stochastic Differential Delay Systems Driven by G-Brownian Motion

This paper focuses on the quasi-sure exponential stability of the stochastic differential delay equation driven by G-Brownian motion (SDDE-GBM): d xi(t)=f(t,xi(t-kappa(1)(t)))dt+g(t,xi(t-kappa(2)(t)))dZ(t), where kappa(1)(<middle dot>), kappa(2)(<middle dot>):R+->[0,tau] denote variable delays, and Z(t) denotes scalar G-Brownian motion, which has a symmetry distribution. It is shown that the SDDE-GBM is quasi-surely exponentially stable for each tau>0 bounded by tau*, where the corresponding (non-delay) stochastic differential equation driven by G-Bronwian motion (SDE-GBM), d eta(t)=f(t,eta(t))dt+g(t,eta(t))dZ(t), is quasi-surely exponentially stable. Moreover, by solving the non-linear equation on tau, we can obtain the implicit lower bound tau*. Finally, illustrating examples are provided.