Journal of Applied Mathematics

Volume 2010, Article ID 425762, 26 pages

http://dx.doi.org/10.1155/2010/425762

## Stability of Nonlinear Neutral Stochastic Functional Differential Equations

^{1}School of Management, Huazhong University of Science and Technology, Wuhan 430074, China^{2}School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received 17 June 2010; Revised 14 September 2010; Accepted 18 September 2010

Academic Editor: Neville Ford

Copyright © 2010 Minggao Xue et al. 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

Neutral stochastic functional differential equations (NSFDEs) have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.

#### 1. Introduction

Stochastic modelling has played an important role in many areas of science and engineering for a long time. Some of the most frequent and most important stochastic models used when dynamical systems not only depend on present and past states but also involve derivatives with functionals are described by the following neutral stochastic functional differential equation: The conditions imposed on their studies are the standard uniform Lipschitz condition and the linear growth condition. The classical result is described by the following well-known Mao's test see [1, page 202, Theorem ].

Theorem 1.1. *Assume that there exist positive constants , and such that**
for all Then there exists a unique solution to (1.1) with initial data (i.e., is an measurable -valued random variable such that ).*

Theorem 1.1 requires that the coefficients and satisfy the Lipschitz condition and the linear growth condition. However, there are many NSFDEs that do not satisfy the linear growth condition. For example, the following nonlinear NSFDE: where coefficients and do not obey the linear growth condition although they are Lipschitz continuous. To the authors' best knowledge, there is so far no result that shows that (1.3) has a unique global solution for any initial data.

On the other hand, we still encounter a new problem when we attempt to deduce the exponential decay of the solution even if there is no problem with the existence of the solution. For example, Mao [2] initiated the following study of exponential stability for NSFDEs employing the Razumikhin technique.

Theorem 1.2. *Let be all positive numbers and for any **
and assume that there exists a function such that
**
for all and also for all **
provided satisfying
**
for all Then for all **
where *

It is very difficult to verify the conditions of Theorem 1.2, and it is clear that does not hold for many NSFDEs. In fact, for (1.3), if one chooses then Here, the polynomial appears on the right-hand side, and it has an order of 4 which is higher than the order of More recently, Mao [3–5], Zhou et al. [6, 7], Yue et al. [8] and Shen et al. [9] provided with some useful criteria on the exponential stability employing the Lyapunov function, but their tests encounters the same problem.

Therefore, we see that there is a necessity to develop new criteria for NSFDEs where the linear growth condition may not hold while the bound on the operator may take a much more general form. In the paper, we will establish a Khasminskii-type test for NSFDEs that cover a wide class of highly nonlinear NSFDEs referring to Khasminskii-type theorems [10] and Mao and Rassias [11] results of stochastic delay differential equations. To our best knowledge, there is no such result for NSFDEs and stochastic functional differential equations (SFDEs).

In the next section, we will establish a general existence and uniqueness theorem of the global solution to (1.1) after giving some necessary notations. Boundedness and Moment stability are given under the Khasminskii-type condition in Section 3. Section 4 establishes asymptotic stability theorem by using semimartingale convergence theory. Section 5 gives corresponding criteria for stochastic functional differential equations. Finally, several examples are given to illustrate our results.

#### 2. Global Solution of NSFDEs

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 an *m*-dimensional continuous local martingale with defined on the probability space. If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by , while its operator norm is denoted by (without any confusion with ). denote the family of all continuous functions from to with the norm where is the Euclidean norm in Denoted by the family of all bounded, -measurable, -valued random variables.

Consider an *n*-dimensional neutral stochastic functional differential equation
on with initial data and
are Borel measurable. Let denote the solution of (2.1) while which is regarded as a -valued stochastic process, denoted by

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

For the purpose of stability, assume that This implies that (2.1) admits a trivial solution, Furthermore, we impose the following assumptions. (H1) (The local Lipschitz condition). For each integer there is a positive constant such that for those with and (H2) There exists a positive constant and a probability measure such that for any (H3) There are two functions and as well as positive constants and a probability measure on such that for all

*Remark 2.1. *In condition (2.7), we see that the function plays a key role in allowing coefficients and to be nonlinear functions.

Theorem 2.2. *Assume that (H1), (H2), and (H3) hold. Then for any initial condition there exists a unique global solution to (2.1) on Moreover, the solution has the properties that**
for any *

*Proof. *It is clear that for any initial data there exists a unique maximal local solution on where is the explosion time [1], by applying the standing truncation technique (see Mao [12, 13]) to (2.1). According to (H2), we have

Let be sufficiently large such that Define the stopping time where throughout this paper, we set denotes the empty sets). Clearly, is increasing as Denote a.s. We will show that a.s., which implies that is global.

Itô formula and condition (2.7) yield for For any and we integrate both sides of (2.12) from 0 to and then take the expectations to get According to the integral substitution technique, we estimate Similiarly, Substituting for (2.14) and (2.15) into (2.13), and by using the Fubini theorem, the result is where Equations (2.6) and (2.9) imply thus, is a finite constant. By using inequality thus, condition (2.6) yields (H2) and the Hölder inequality yield Substituting for (2.16), (2.18), and (2.19) into (2.17), the result is For any (2.20) implies Let then Therefore, for any By (2.6), we may obtain For any the Gronwall inequality implies Thus, for all which implies Since defining for according to (2.26), then Clearly, condition (2.6) implies Letting in (2.28), then namely, Moreover, setting in (2.16), we may obtain that that is, Let us now proceed to prove a.s. given that we have shown (2.27)–(2.31). For any and we can integrate both sides of (2.12) from 0 to and then take expectations to get where By the Gronwall inequality and (2.32), we have In particular, This implies Letting , by (2.6), then that is, By (2.32), we may obtain that that is, Repeating this procedure, we can show that, for any integer a.s. and and where We must therefore have a.s. as well as the required assertion.

Note that condition (2.6) may be replaced by more general condition which is suitable to the corresponding results below.

#### 3. Boundedness and Moment Stability

In the previous section, we have shown that the solution of (2.1) has the properties that for any In the following, we will give more precise estimations under specified conditions; that is, we will establish the criteria of moment stability and asymptotic stability of the solution to (2.1) under specified conditions.

Theorem 3.1. *Assume that (H1), (H2), and (H3) hold except (2.7) which is replaced by**
for all where are constants and and are probability measures on Then for any initial data , the global solution to (2.1) has the property that
**
where while and are the unique roots to the following equations:
**
respectively. If then
*

*Proof. *We first observe that (3.2) implies (2.7) if we set and So, for any initial data, (2.1) has a unique global solution on which has the properties (2.8). Based on these properties, we can apply the Itô formula and condition (3.2) to obtain that for any
We integrate both sides of the above inequality from to and take expectations to get
by using of Compute
Similiarly,
Substituting for (3.8) and (3.9) into (3.7), the result is
where It is clear that, for we have hence,
By (H2) and (H3) and inequality we compute
For any
Leting since then
and by (H3), then
Therefore,
When then that is,
On the other hand, when by (3.7) and the Itô formula, we may show easily that
By and the Fubini theorem, we obtain
The proof is complete.

#### 4. Asymptotic Stability

In this section, we will establish asymptotic stability of (2.1) without the linear growth condition. It is well known that the linear growth condition is one of the most important conditions to guarantee asymptotic stability. Therefore we introduce the following semitingale convergence thoerem [14, 15], which will play a key role in dealing with nonlinear systems.

Lemma 4.1. *Let be a real-valued local martingale with a.s. Let be a nonnegative -measurable random variable. If is a nonnegative continuous -adapted process and satisfies for then is almost surely bounded, namely, a.s.*

Theorem 4.2. *Assume that (H1), (H2), and (H3) hold except (2.7) which is replaced by**
for all where Then, for any initial data, the unique global solution of (2.1) has the property that
**
where while and are the unique roots to the following equations:
**
respectively.*

*Proof. *We first observe that (4.1) implies (2.7) if we set and So, for any initial data, (2.1) has a unique global solution on which has the properties (2.8). Similar to the proof of Theorem 3.1, applying the Itô formula and condition (4.1), for any we may obtain that
For we can integrate both sides of the above inequality from to and take expectations to get
where is a real-valued continuous local martingale with Similar to Theorem 3.1, we have
Lemma 4.1 implies
Since then
According to the definition of , we compute
Therefore, we may also compute
Noting that choose Then and we obtain
(4.8) and (4.11) yield
Recall the condition which implies The required result is obtained.

*Remark 4.3. *From the processes of the proof of Theorems 3.1 and 4.2, we see that condition (2.6) plays an important role in dealing with the neutral term. Moreover, applying condition (2.6), we can also obtain more precise results
In the next section, condition (2.6) will be replaced by a more general condition for stochastic functional differential equation.

#### 5. Stochastic Functional Differential Equation

Let . Then (2.1) reduces to This is a stochastic functional differential equation. In this section, we will give the corresponding results for stochastic functional differential equation. We will also see that the conditions are more general.

Define an operator from to by We impose the following assumption which is more general than (H3).(H) There are two functions and as well as two positive constants and a probability measure on such that for all

Theorem 5.1. *Assume that (H1) and (H) hold. Then for any initial condition there exists a unique global solution of (5.1) on Moreover, the solution has the properties that**
for any *

*Proof. *Since the proof is similar to Theorem 2.2, we will only outline the proof. It is clear that for any initial data there is a unique maximal local solution on where is the explosion time [1]. Let be sufficiently large for
Define the stopping time
where throughout this paper, we set . Clearly is increasing as Denote a.s. We will show that a.s., which implies a.s. By Itô formula and (5.4), for any and we obtain
where For any the Gronwall inequality yields
which implies
Defining for according to (5.3), then
Condition (5.3) implies Letting in (5.11), then namely,
Moreover, setting in (5.8), we may obtain that
that is,
Let us now proceed to prove a.s. given that we have shown (5.10)–(5.14). For any and we get
where
By the Gronwall inequality and (5.8), we have
In particular,