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

Shounian Deng; Chen Fei; Weiyin Fei; Xuerong Mao

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

Shounian Deng, Chen Fei, Weiyin Fei, Xuerong Mao

Mathematics And Statistics

Research output: Contribution to journal › Article

Abstract

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.

Language	English
Journal	Applied Mathematics Letters
Publication status	Accepted/In press - 23 Apr 2019

Fingerprint

Euler-Maruyama Method

Brownian movement

Delay Equations

Lyapunov functions

Asymptotic stability

Stochastic Equations

Brownian motion

Stochastic Differential Delay Equations

Equivalence

Lipschitz condition

Computer simulation

Exponential Stability

Lyapunov Function

If and only if

Numerical Simulation

Numerical Examples

Keywords

mean square stability
G-SDDE
EM method
stability equivalence
G-Simulation
G-Brownian motion
Euler-Maruyama method
stochastic differential delay equations

Cite this

Deng, S., Fei, C., Fei, W., & Mao, X. (Accepted/In press). Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method. Applied Mathematics Letters.

Deng, Shounian ; Fei, Chen ; Fei, Weiyin ; Mao, Xuerong. / Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method. In: Applied Mathematics Letters. 2019.

@article{f963ac813f2e4a679438549103355e17,

title = "Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method",

abstract = "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. ",

keywords = "mean square stability, G-SDDE, EM method, stability equivalence, G-Simulation, G-Brownian motion, Euler-Maruyama method, stochastic differential delay equations",

author = "Shounian Deng and Chen Fei and Weiyin Fei and Xuerong Mao",

year = "2019",

month = "4",

day = "23",

language = "English",

journal = "Applied Mathematics Letters",

issn = "0893-9659",

}

Deng, S, Fei, C, Fei, W & Mao, X 2019, 'Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method' Applied Mathematics Letters.

Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method. / Deng, Shounian; Fei, Chen; Fei, Weiyin; Mao, Xuerong.

In: Applied Mathematics Letters, 23.04.2019.

Research output: Contribution to journal › Article

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

PY - 2019/4/23

Y1 - 2019/4/23

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

M3 - Article

JO - Applied Mathematics Letters

T2 - Applied Mathematics Letters

JF - Applied Mathematics Letters

SN - 0893-9659

ER -

Deng S, Fei C, Fei W, Mao X. Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method. Applied Mathematics Letters. 2019 Apr 23.