混杂随机微分方程θ方法的几乎必然指数稳定性
Almost sure exponential stability of θ-method for hybrid stochastic differential equations

大部分的混杂随机微分方程很难得到解析解, 因此利用数值方法研究其数值解具有重要意义. 本文研究θ方法产生的数值解的几乎必然指数稳定性. 在单边Lipschitz条件和线性增长条件下, 首先给出方程的平凡解是几乎必然指数稳定的. 然后在相同条件下, 运用Chebyshev不等式和Borel-Cantelli引理, 证明了对θ ∈ [0,1], θ方法重现平凡解的几乎必然指数稳定性. θ方法是一种比现有的Euler-Maruyama方法和向后Euler-Maruyama方法更广的方法. 当θ等于1或0时,它分别退化为上述两种方法之一. 本文的结论对上述两种方法同样适用. 最后, 数值例子和仿真说明了对不同的θ所提出方法的有效性和稳定性.
It is difficult to obtain analytical solutions for most of the hybrid stochastic differential equations (SDEs), so the research on the numerical solutions by the use of numerical methods is of great significance. This paper focuses on the almost sure exponential stability of the numerical solutions produced by the θ-method. Under the one-sided Lipschitz condition and the linear growth condition, the almost sure exponential stability of the trivial solution for hybrid SDEs is first introduced. Then, by applying the Chebyshev inequality and the Borel-Cantelli lemma, we prove that the θ-method reproduces the corresponding stability of the trivial solution under the same conditions for θ ∈ [0,1]. The θ-method is a more general method than the existing Euler-Maruyama method as well as the backward Euler-Maruyama method. When θ isequal to 1 or 0, it degenerates to one of the above twomethods, respectively. The results of this paper are also applicable to these two methods. Finally, a numerical example and its simulations with different θ are given to illustrate the effectiveness and the stability of the proposed method.