Stochastic Functional Differential Equation under Regime Switching

Ling Bai,Zhang Kai

doi:10.1155/2012/206438

Abstract

We discuss stochastic functional differential equation under regime switching dx(t) = f(xt, r(t), t)dt + q(r(t))x(t)dW1(t) + σ(r(t)) | x(t)|βx(t)dW2(t). 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.

Highlights

Using the ergodic property of the Markov chain, we give the sufficient condition of almost surely exponentially stable of this system. 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
In this paper we will consider the following stochastic functional equation: dx t f xt, r t, t dt qrtxt dW1 t σ r t |x t |βx t dW2 t. The switching between these N regimes is governed by a Markovian chain r t on the state space S {1, 2, . . . , N}. xt ∈ C −τ, 0 ; Rn is defined by xt θ xt θ;θ ∈
We should emphasize that 1, Page 305 the operator V thought as a single notation rather than acting on V is defined on C −τ, 0 ; Rn × R × S V is defined on Rn × −τ, ∞ × S

Summary

Introduction

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. In this paper we will consider the following stochastic functional equation: dx t f xt, r t , t dt qrtxt dW1 t σ r t |x t |βx t dW2 t. The switching between these N regimes is governed by a Markovian chain r t on the state space S {1, 2, . Throughout this paper, let C2,1 Rn × −τ, ∞ × S; R denote the family of all positive real-valued functions V x, t, k on Rn × −τ, ∞ × S which are continuously twice differentiable in x and once in t. We should emphasize that 1, Page 305 the operator V thought as a single notation rather than acting on V is defined on C −τ, 0 ; Rn × R × S V is defined on Rn × −τ, ∞ × S

Global Solution

Asymptotic Boundedness

Stabilization of Noise