TY - JOUR

T1 - Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method

AU - Deng, Shounian

AU - Fei, Chen

AU - Fei, Weiyin

AU - Mao, Xuerong

N1 - © 2019 Elsevier Ltd.
Shounian Deng, Chen Fei, Weiyin Fei, Xuerong Mao, Stability equivalence between the stochastic differential delay equations driven by G-Brownian motion and the Euler–Maruyama method, Applied Mathematics Letters, Volume 96, 2019, Pages 138-146, https://doi.org/10.1016/j.aml.2019.04.022

PY - 2019/10/1

Y1 - 2019/10/1

N2 - Consider a stochastic differential delay equation driven by G-Brownian motion (G-SDDE)
dx(t) = f(x(t), x(t − τ))dt + g(x(t), x(t − τ))dB(t) + h(x(t), x(t − τ))dhBi(t).
Under the global Lipschitz condition for the G-SDDE, we show that the G-SDDE is exponentially stable in mean
square if and only if for sufficiently small step size, the Euler-Maruyama (EM) method is exponentially stable in
mean square. Thus, we can carry out careful numerical simulations to investigate the exponential stability of the
underlying G-SDDE in practice, in the absence of an appropriate Lyapunov function. A numerical example is
provided to illustrate our results.

AB - Consider a stochastic differential delay equation driven by G-Brownian motion (G-SDDE)
dx(t) = f(x(t), x(t − τ))dt + g(x(t), x(t − τ))dB(t) + h(x(t), x(t − τ))dhBi(t).
Under the global Lipschitz condition for the G-SDDE, we show that the G-SDDE is exponentially stable in mean
square if and only if for sufficiently small step size, the Euler-Maruyama (EM) method is exponentially stable in
mean square. Thus, we can carry out careful numerical simulations to investigate the exponential stability of the
underlying G-SDDE in practice, in the absence of an appropriate Lyapunov function. A numerical example is
provided to illustrate our results.

KW - mean square stability

KW - G-SDDE

KW - EM method

KW - stability equivalence

KW - G-Simulation

KW - G-Brownian motion

KW - Euler-Maruyama method

KW - stochastic differential delay equations

U2 - 10.1016/j.aml.2019.04.022

DO - 10.1016/j.aml.2019.04.022

M3 - Article

SN - 0893-9659

VL - 96

SP - 138

EP - 146

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

ER -