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

Junhao Hu and Zhiying Xu

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.

1. Introduction

In many practical dynamical systems such as neural networks, computer aided design, population ecology, chemical process simulation, and automatic control, stochastic differential equations represent the class of important dynamics (see [1–4]). During the recent several years, the asymptotic properties of neutral stochastic functional differential equations have been investigated by many authors (see [5–14]). Mao [10, 11] gave the exponential stability of neutral stochastic functional differential equations by using the Razumikhin-type theorems. Zhou and Hu [14] used the same argument to discuss the exponential stability in th moment of neutral stochastic functional differential equations and neutral stochastic functional differential equations with Markovian switching. Wu et al. [13] examined the almost sure robust stability of nonlinear neutral stochastic functional differential equations with infinite delay, including the exponential stability and the polynomial stability. Song and Shen [12] investigated the asymptotic behavior of neutral stochastic functional differential equations under the more general conditions than the classical linear growth condition. Chen et al. [5] considered the exponential stability in mean square moment of mild solution for impulsive neutral stochastic partial functional differential equations by employing the inequality technique. The attraction and quasi-invariant sets of neutral stochastic partial functional differential equations were also studied in the recent paper [9].

In this paper, we will consider neutral stochastic functional differential equations with two-time-scale Markovian switching modeled by a continuous-time Markov chain which has a large state space. The computational effort and the complexity become a real concern. To overcome the difficulties, we have devoted much effort to the modeling and analysis of such systems, in which one of the main ideas is to split a large-scale system into several classes and lump the states in each class into one state (see [3, 15–21]). Khasminskii et al. for the first time established the asymptotic properties of the Markov chain by introducing a small parameter (see [22]). Yin and Zhang developed the method in their book [4] that a complicated system can be replaced by the corresponding limit system that has a much simpler structure. Motivated by the papers [16, 21], under suitable conditions, using the stability of the limit system as a bridge, we will study the exponential stability of neutral stochastic functional differential equations with Brownian motion and Poisson jump by utilizing perturbed Lyapunov function methods and Razumikhin-type criteria.

The remainder of this paper is organized as follows. In Section 2, we introduce some notations and notions needed in our investigation. In Section 3, we state our main results, that is, exponential stability of neutral stochastic functional differential equations with two-time-scale Markovian switching. The exponential stability for neutral stochastic functional differential equations driven by pure jumps is also discussed in Section 4. Finally, two examples are presented to justify and illustrate applications of the theory in Section 5.

2. Preliminaries

Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous and contains all -null sets). Let be an -dimensional Brownian motion defined on the probability space. For , let denote the family of continuous functions from to with norm , where is the Euclidean norm in . If is a vector or matrix, its transpose is denoted by , while its trace norm is denoted by . Denote by the family of all measurable and bounded -valued random variables. For and , denote by the family of all -measurable -valued random variables such that . We will denote the indicator function of a set by .

Consider an -dimensional neutral stochastic functional differential equation with Markovian switching as follows:

on with initial data and , which is regarded as a -valued stochastic process. Moreover, , , .

Let () be a right-continuous Markov chain on the probability space taking values in a finite state space with generator given by

where . Here, is the transition rate from to if , while .

We assume the Markov is independent of the Brownian motion . It is well known that almost every sample path is a right-continuous step function with finite number of simple jumps in any finite subinterval of . As a standing hypothesis, we assume that the Markov chain is irreducible. This is equivalent to the condition that, for any , we can find , such that

Then, always has an eigenvalue . The algebraic interpretation of irreducibility is rank . Under this condition, the Markov chain has a unique stationary distribution , subject to and for all . For a real-valued function defined on , we define

for each .

Let denote the family of all nonnegative functions on , which are continuously twice differentiable in and once differentiable in . If , define an operator from to by

where

For a parameter , we rewrite the Markov chain as and the generator as . is given by

where represents the fast varying motions and represents the slowly changing dynamics. Set , , and . For the sake of simplicity, suppose that

with , , and

where is a generator of a Markov chain taking values in , for every .

We give the first assumption as follows.

Assumption 1. For each , is irreducible.

In order to emphasize the effect of the fast switching, (1) can be given by

To assure the existence and uniqueness of the solution, we give the following standard assumptions.

Assumption 2 (local Lipschitz condition). For each integer , there exists a constant such that

for all , and those with , and , .

Assumption 3 (linear growth condition). There is an , for any , , such that

Assumption 4. For all and those , there is a constant such that

Under Assumptions 2, 3, and 4, (10) has a unique solution denoted by on , where is dependent on the initial value (see [8]). Moreover, for every and any compact subset of , there is a positive constant which is independent of such that

Since the state space of the Markov chain is large, it is too complicated to deal with directly. We need to analyse the limit equation of (10). To continue, make all the states in each into a single state and define an aggregated process as

Denote the state space of by , the stationary distribution by , and . Define

with and , . It has been known that converges weakly to as , where is a continuous-time Markov chain with generator and state space (see [4]). Define

for each with and . It is easy to know that , , and are the limits with respect to the stationary distribution of the Markov chain. Consider that, for any , are nonnegative definite matrices, so we denote its “square root” of by . For degenerate diffusions, we can see the argument in [23].

The limit equation of (10) is defined as follows:

3. Exponential Stability of NSFDE with Two-Time-Scale Markovian Switching

In this section, we establish the Razumikhin-type theorem on the exponential stability for (10). Denote by the family of nonnegative real-valued functions defined on that are -times continuously differentiable with respect to . At the same time, we need another assumption and a lemma with respect to for some .

Assumption 5. For each , as . Moreover, , for , where denotes the th derivative of with respect to and denotes the function of satisfying .

Lemma 6. Suppose that ; there is a positive constant such that

Then, for any , the solution for (10) satisfies

Proof. Note the following elementary inequality:

We have from condition (20) that, for any ,

Then,

Therefore, the desired result holds.

Theorem 7. Let Assumptions 1–4 hold and let , , , be all positive numbers and . Assume that there exists a function satisfying Assumption 5, such that

for all , , . Consider the following:

for all , , , and

provided , satisfying

for all . Then, for all ,

where

being the root of the following equation:

In other words, the trivial solution of (10) is th moment exponentially stable and the th moment Lyapunov exponent is not greater than .

Proof. Let

By the definition of , we know that

Extend to by setting . Recalling the facts that is continuous for all and is right continuous, it is easy to see that is right continuous on . Let be arbitrary and define

for all . We claim that

Note that, for each , either or .

If , because is right continuous on , it is easy to obtain that, for all sufficiently small, ; hence, and .

If , we have

for all .

Then,

for all .

On the other hand, by Lemma 6, we derive

Then,

where ; that is, .

Consequently, there exists a sufficiently small , such that, for any ,

for all . Thus,

which implies that

By the condition of , we get

Next, we consider

By the definition of operator , we have

Therefore,

By the definition of ,

This implies that

By the argument of Lemma  7.14 in [4], the right side of the above inequality is equivalent to ; that is, . Similarly, we can show that

By the definition of and , we have

Hence,

By the argument of Lemma  7.14 in [4], the right side of the above inequality is equivalent to ; that is, . Therefore,

That is

So, for all sufficiently small, and hence . Inequality (34) holds.

It follows from (34) that for all . By the definition of ,

By Lemma 6, we derive

That is,

4. Neutral Stochastic Functional System with Pure Jump

In this section, we discuss the stability of the following neutral stochastic functional system with pure jump:

where , , . We assume that each column of the matrix depends on only through the th coordinate ; that is,

is an -dimensional Poisson process and the compensated Poisson process is defined by

where are independent one-dimensional Poisson random measures with characteristic measure coming from independent one-dimensional Poisson point processes. The limit system of (56) is defined as follows:

where and . Similar to the definition of , we define

for each with and .

To assure the existence and uniqueness of the solution of (59), we also give the following standard assumptions.

Assumption 8. For any integer , there is a constant , such that

for all and those with , and, .

Assumption 9. There is an , such that, for any , ,

Assumption 10. For all and those , there is a constant such that

Given that , define an operator by

where

Lemma 11 (see [20]). Let Assumptions 1, 8, and 9 hold, as ; then, converges weakly to in , where is the space of functions defined on that are right continuous and have left limits taking values in and are endowed with the Skorohod topology.

Theorem 12. Let Assumptions 1 and 8–10 hold and let , , , be all positive numbers and . Assume that there exists a function satisfying Assumption 5, such that

for all and . Consider the following:

for all , , , and

provided , satisfying

Then, for all , ,

where

being the root of the following equation:

Proof. Define

Extend to by setting ; then, is right continuous on . Let be arbitrary and define

for all . We claim that

Similar to the proof of Theorem 7, we derive

for all , where ; that is, .

Thus,

which implies that

By the condition of , we get

We now consider

By the definition of the operator , we have

This implies that

By the definition of ,

By Assumption 8, we have

By the argument of Lemma  7.14 in [4], the right side of the above inequality is equivalent to ; that is, . Similarly, by mean-value theorem, we can show that there exists which is between and such that

By the argument of Lemma  7.14 in [4], we have . Similar to the proof of Theorem 7, we derive , . Therefore, we arrive at

Then,

Similar to the proof of Theorem 7, we get

The proof is therefore completed.

5. Examples

We will give two examples to illustrate our theory.

Example 1. Let be a Markov chain generated by given in (14) with

The generator is made up of two irreducible blocks; by

and , we get . In the same way, by

and , we have . So,

Consider a one-dimensional neutral stochastic functional differential equation as follows:

with

For any and , applying the Hölder inequality yields

which implies condition (24). Then, the limit equation is

where is the Markov chain generated by and

We define , . And by simple calculation, we can get

Consequently,

It is easy to find a such that . Therefore, for any satisfying on , (100) yields

Hence, by Theorem 7, the solution is mean square stable when is sufficiently small.

Example 2. Let be a Markov chain generated by

Here, we set . By a similar way, we get the stationary distribution .

Consider the following one-dimensional equation:

with

Let

For any and , applying the Hölder inequality yields

which implies condition (67). Then, the limit equation is

Let ; then

We can find a such that . Therefore, for any satisfying on , (108) yields

Hence, by Theorem 12, the solution is mean square stable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was supported by the National Science Foundation of China with Grant no. 61374085.

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