On p th moment exponential stability of stochastic differential equations with Markovian switching and time-varying delay

Enwen Zhu,Xue Tian,Yueheng Wang

doi:10.1186/s13660-015-0657-9

Abstract

In this paper we discuss the problem of pth moment exponential stability for general nonlinear stochastic differential equations with Markovian switching and time-varying delay. By using the Lyapunov function, the stochastic analysis technique and the generalized Halanay inequality, some novel sufficient conditions on pth moment exponential stability of stochastic differential equations with Markovian switching and time-varying delay are derived. The results obtained in this paper are completely new and modify and improve some known results. Moreover, two numerical examples are also provided to demonstrate the effectiveness and applicability of the theoretical results. The aim of this paper is to investigate pth moment exponential stability of general nonlinear stochastic differential equations with Markovian switching and time-varying delay.

Highlights

1 Introduction In the last decades, the stability of stochastic differential equations has been extensively studied by many authors
Motivated by the above discussions, the purpose of this paper is to study pth moment exponential stability of a general nonlinear stochastic differential equation with Markovian switching and time-varying delay
5 Conclusions In this paper, we have investigated pth moment exponential stability of stochastic differential equations with Markovian switching and time-varying delay

Summary

Introduction

The stability of stochastic differential equations has been extensively studied by many authors (see, e.g., [ – ]). The stability of stochastic differential equations with Markovian switching (or jumping) has received a lot of attention. Ji and Chizeck [ ] and Mariton [ ] studied the stability of a jump linear equation. Basak et al [ ] discussed the stability of a semi-linear stochastic differential equation with Markovian switching. Mao [ ] as well as Shaikhet [ ] investigated the stability for nonlinear stochastic differential delay equations with Markovian switching. For a systematic and detailed review about stochastic differential equations with Markovian switching, Mao and Yuan [ ] should be referred to

Objectives

Conclusion