Author(s): MO Hao-yi, DENG Fei-qi, PENG Yun-jian, Systems Engineering Institute, South China University of Technology, School of Applied Mathematics, Guangdong University of Technology
Pages: 1246-1253
Year: 2015 Issue:  9
Journal: Control Theory & Applications
Keyword:  Brownian motion; θ-method; Markov chains; almost sure exponential stability; hybrid systems;
Abstract: 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 θis equal to 1 or 0,it degenerates to one of the above two methods,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.