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

Research output: Contribution to journalArticle

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. 
LanguageEnglish
JournalApplied Mathematics Letters
Publication statusAccepted/In press - 23 Apr 2019

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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

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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",
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language = "English",
journal = "Applied Mathematics Letters",
issn = "0893-9659",

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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 -