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 -