ALMOST SURE EXPONENTIAL STABILITY IN THE NUMERICAL SIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
ALMOST SURE EXPONENTIAL STABILITY IN THE NUMERICAL SIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
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DOI：
10.1137/140966198
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发表时间：
2015-01-01
影响因子：
2.9
通讯作者：
Mao, Xuerong
中科院分区：
文献类型：
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作者：
Mao, Xuerong
本文主要研究随机微分方程的指数稳定性是否与数值方法的指数稳定性相同。在全局Lipschitz条件下，我们首先证明SDE是第p阶矩指数稳定的（因为p是（0,1）的一个元素），当且仅当随机θ方法在足够小的步长下是第p阶矩指数稳定的。然后，我们证明了SDE或随机θ方法的第p阶矩指数稳定性分别暗示了SDE或随机θ方法的几乎肯定的指数稳定性。因此，我们的新理论使我们能够使用随机theta方法来研究SDEs的几乎肯定的指数稳定性，而不是Lyapunov函数的方法。也就是说，我们现在可以使用足够小的步长δ t的随机θ方法进行仔细的数值模拟。如果随机θ方法在足够小的p（0,1）下是第p阶矩指数稳定的，那么我们可以推断基本的SDE几乎肯定是指数稳定的。我们的新理论也使我们能够展示随机θ方法重现SDEs几乎肯定的指数稳定性的能力。特别地，我们给出了第1节中列出的（P1）和（P2）两个开放问题的正答案。
This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschitz condition, we first show that the SDE is pth moment exponentially stable (for p is an element of(0, 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size Delta t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p. (0, 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems, (P1) and (P2) listed in section 1.