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

Deng, Shounian and Fei, Chen and Fei, Weiyin and Mao, Xuerong (2019) Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method. Applied Mathematics Letters. ISSN 0893-9659 (In Press)

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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.