Exponential Stability Of Two Classes Of Stochastic Differential Equations Driven By G-Brownian Motion

In this paper,we discuss stability of two classes of the stochastic differential equa-tions driven by G-Brownian motion.This work is divided into two Sections.In the first Section,we consider stochastic functional differential equations with infinite delay driven by G-Brownian motion(G-ISFDEs,in short):dy(t)=f(t,yt)dt+h(t,yt)d(B)(t)+σ(t,yt)dB(t),t≥0,(1)where yt=y(t+θ):={y(t+θ):-∞<θ≤0)} for t≥0,f:R+× BC((-∞,0];Rn)→Rn,:R+×SC((-∞,0];Rn)→Rn anda:R+×BC((-∞,0];Rn)→Rn,BC((-∞,0];Rn)denotes the family of bounded continuous Rn-valued function φ defined on(-∞,0]with norm ‖φ‖=supθ≤0|φ(θ)|,B(·)is a G-Brownian motion,(B)(·)is the quadratic variation process of the G-Brownian motion B(·).We prove the existence and uniqueness of the solution for G-ISFDEs,where the coefficients satisfy local Lipschitz and Lyapunov-typo conditions.At the same time,we propose the sufficient conditions on the p-th momen-t exponential stability of the trivial solution for G-ISFDEs by virtue of G-Lyapunov function method.Motivated by the proofs in Section 1.we give the existence and uniqueness of the solution for pantograph stochastic differential equations driven by G-Brownian motion(G-PSDEs,in short)in Section 2 as follows:dy(t)=f(t,y(t),y(θt))dt+h(t,y(t),y(θt))d(B)(t)+σ(t,y(t),y(θdt))dB(t),t≥0,(2)here y(0)=ξ∈Rn is the initial value,0<θ<1,f:R+× Rn×Rn→Rn,h:R+×Rn×Rn→Rn anda:R+× Rn × Rn→Rn,B(·)is a G-Brownian motion,(B)(·)is the corresponding quadratic variation process.At the same time,the asymptotic boundedness and exponential stability of for G-PSDEs are derived.