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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 907982, 15 pages
http://dx.doi.org/10.1155/2014/907982
Research Article

Exponential Stability of Neutral Stochastic Functional Differential Equations with Two-Time-Scale Markovian Switching

College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China

Received 22 December 2013; Accepted 16 January 2014; Published 16 March 2014

Academic Editor: Weihai Zhang

Copyright © 2014 Junhao Hu and Zhiying Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We develop exponential stability of neutral stochastic functional differential equations with two-time-scale Markovian switching modeled by a continuous-time Markov chain which has a large state space. To overcome the computational effort and the complexity, we split the large-scale system into several classes and lump the states in each class into one class by the different states of changes of the subsystems; then, we give a limit system to effectively “replace” the large-scale system. Under suitable conditions, using the stability of the limit system as a bridge, the desired asymptotic properties of the large-scale system with Brownian motion and Poisson jump are obtained by utilizing perturbed Lyapunov function methods and Razumikhin-type criteria. Two examples are provided to demonstrate our results.