Stability equivalence between the stochastic differential delay equations driven by G-Brownian motion and the Euler-Maruyama method
Stability equivalence between the stochastic differential delay equations driven by G-Brownian motion and the Euler-Maruyama method
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DOI：
10.1016/j.aml.2019.04.022
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
Shounian Deng;Chen Fei;W. Fei;X. Mao
中科院分区：
文献类型：
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作者：
Shounian Deng;Chen Fei;W. Fei;X. Mao
考虑由G-布朗运动驱动的随机时滞微分方程（G-SDDE）dx（t）= f（x（t），x（t− τ））dt + g（x（t），x（t− τ））d B（t）+ h（x（t），x（t− τ））d（t）.在全局Lipschitz条件下，证明了G-SDDE是均方指数稳定的当且仅当对于足够小的步长，Euler-Maruyama（EM）方法是均方指数稳定的.因此，我们可以进行仔细的数值模拟，调查指数稳定性的基础G-SDDE在实践中，在没有一个适当的李雅普诺夫函数。最后给出了一个数值例子来说明我们的结果。
Consider a stochastic differential delay equation driven by G-Brownian motion (G-SDDE) d x (t)= f (x (t), x (t− τ)) d t+ g (x (t), x (t− τ)) d B (t)+ h (x (t), x (t− τ)) d(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.