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

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

Keywords

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

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