Stochastic Functional Differential Equation under Regime Switching

We discuss stochastic functional differential equation under regime switching . We obtain unique global solution of this system without the linear growth condition; furthermore, we prove its asymptotic ultimate boundedness. Using the ergodic property of the Markov chain, we give the sufficient condition of almost surely exponentially stable of this system. 1. Introduction Recently, many papers devoted their attention to the hybrid system, they concerned that how to change if the system undergoes the environmental noise and the regime switching. For the detailed understanding of this subject, [1] is good reference. In this paper we will consider the following stochastic functional equation: The switching between these regimes is governed by a Markovian chain on the state space . is defined by ; . denote the family of continuous functions from to , which is a Banach space with the norm . satisfies local Lipschitz condition as follows. Assumption A. For each integer , there is a positive number such that for all and those with . Throughout this paper, unless otherwise specified, we let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all P-null sets). Let ( ), , be the standard Brownian motion defined on this probability space. We also denote by . Let be a right-continuous Markov chain on the probability space taking values in a finite state space with the generator given by where . Here is the transition rate from to and if while We assume that the Markov chain is independent on the Brownian motion ; furthermore, and are independent. In addition, throughout this paper, let denote the family of all positive real-valued functions on which are continuously twice differentiable in and once in . If for the following equation there exists , define an operator from to by where Here we should emphasize that [1, Page 305] the operator (thought as a single notation rather than acting on ) is defined on although is defined on . 2. Global Solution Firstly, in this paper, we are concerned about that the existence of global solution of stochastic functional differential equation (1.1). In order to have a global solution for any given initial data for a stochastic functional equation, it is usually required to satisfy the local Lipschitz condition and the linear growth condition [1, 2]. In addition, as a generation of linear condition, it is also mentioned in [3, 4] with one-sided linear growth condition. The authors improve the results using polynomial growth condition in [5, 6]. After that,