Asymptotical boundedness and stability for stochastic differential equations with delay driven by G-Brownian motion

Delay‐dependent stability of a class of stochastic delay systems driven by G‐Brownian motion

“…Based on the stability criteria, the corresponding upper bound of time delay can also be obtained. This is different from [16, 18], where the stability conditions are related to Lyapunov function. In addition, we also reveal that the quadratic variation part may play a positive role in the mean square and quasi‐sure exponential stability.…”

“…In this case, condition (11) is always true, and in condition (12) can be omitted because it is from the quadratic variation part. Hence, the G‐SDDS (6) is mean square and quasi‐surely exponentially stable if there exists a positive definite matrix P such that where . Remark 5 In [16, 18], the sufficient conditions for the stability of stochastic systems driven by G‐Brownian motion were obtained by means of G‐Lyapunov function. In this paper, we no longer give the Lyapunov‐type conditions, but get the explicit sufficient conditions for stability by constructing a concrete Lyapunov functional.…”

“…Generally, the time delay is often unavoidable due to uncertain communication environment, so it is required to be taken into consideration for stability analysis of stochastic systems. Yin and Ren [18] discussed asymptotical boundedness and exponential stability of stochastic differential delay systems driven by G‐Brownian motion (G‐SDDSs) by means of G‐Lyapunov function. Deng et al .…”

Khasminskii-type theorems for stochastic differential delay equations driven by G-Brownian motion

“…Ren, Jia, and Sakthivel (2016) discussed the pth moment stability of solutions to impulsive G-SDEs. Moreover, some other important properties of G-SDEs have been investigated by many researchers (see Deng, Fei, Fei, & Mao, 2019;Faizullah, 2016;In Press;Hu, Lin, & Hima, 2018;Li & Yang, 2018;Luo & Wang, 2014;Ren et al, 2016;Ren, Yin, & Sakthivel, 2018;Yin, Cao, & Ren, 2019;Yin & Ren, 2017). Mao (2002); Mao and Rassias (2005) established a Khasminskii-type test for SDDEs.…”

Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach

“…Since then, this kind of G-SDEs have generated lots of developments. For more details, we refer the reader to Ren et al [18,19,20,21], Yin and Ren [23], Zhang and Chen [24,25] and the references therein. For the updated developments on G-stochastic analysis and G-SDEs, one can see the survey paper by Peng [16].…”