#### Abstract

Stability and boundedness are two of the most important topics in the study of stochastic functional differential equations (SFDEs). This paper mainly discusses the almost sure asymptotic stability and the boundedness of nonlinear SFDEs satisfying the local Lipschitz condition but not the linear growth condition. Here we assume that the coefficients of SFDEs are polynomial or dominated by polynomial functions. We give sufficient criteria on the almost sure asymptotic stability and the boundedness for this kind of nonlinear SFDEs. Some nontrivial examples are provided to illustrate our results.

#### 1. Introduction

Stochastic modeling plays an important role in many branches of sciences and industries. Since Itô introduced his stochastic calculus, stochastic delay or functional differential equations (SDDEs or SFDEs) have been used successfully to model those systems which depend not only on the present history of the state but also on the past ones (see, e.g., [1–5]). Stability and boundedness are two of the most important topics in the study of SDDEs or SFDEs in modern control theory. Many researchers have done a lot of works for these two topics (see, e.g., [6–18]).

In general, a SFDE has the form on with initial data , where and × (the notations used here will be illustrated in Section 2). Most of the existing stability criteria of SFDEs require the coefficients of corresponding systems to satisfy the local Lipschitz condition and the linear growth condition or the one-side linear growth condition (see, e.g., [2, 4, 5]). However, many SDDEs or SFDEs can not be dominated by the linear growth condition, such as stochastic population system, Lotka-Volterra systems, and system (2) as follows: with initial data , where is a scalar Brownian motion, are bounded linear operators from to satisfying , and are probability measures on , .

So it is necessary to consider the cases of the nonlinear growth condition. Recently, Liu et al. [10] study the asymptotic stability of nonlinear stochastic differential equations (SDEs) with polynomial growth condition, and they also develop their results to the case of SDDEs [11]. In this paper, we mainly establish some new results on the almost sure asymptotic stability and the boundedness in the sense of the pth moment and the trajectory with large probability of SFDEs with polynomial growth condition, which imply the results in [10, 11].

Here we would like to mention the work of Luo et al. [12]. It proposes a generalized theory for the asymptotic stability and the boundedness for SFDEs based on a Lyapunov-type condition without the linear growth condition; more precisely, the diffusion operator of a Lyapunov function is required to satisfy the following condition: where and are probability measures on , , , . However as to system (2), setting , we can get Since the above includes the positive term , it does not satisfy (3). So their work does not imply ours. Also Shen et al. [13] use the LaSall technique to study the almost sure asymptotical stability of SFDEs under different settings.

The organization of this paper is as follows: Section 2 describes some necessary notations and lemmas; the existence of the global solution and the bounedness of SFDEs are stated in Section 3; sufficient conditions are proposed for the almost sure asymptotic stability in Section 4; to show the applications of our results, some illustrative examples are given in the final section.

#### 2. Preliminaries

Through this paper, let be a complete probability space with a filtration satisfying the usual conditions and an -dimensional Brownian motion defined on the probability space. Let and let denote the family of all continuous -valued functions on with the norm . Let be the family of all bounded, -measurable, -valued, -adapted stochastic processes. Let be probability measures on , which satisfy (). Let be the family of all functions such that . is a continuous -valued stochastic process on . We assume for all , which is regarded as a -valued stochastic process.

Consider an -dimensional SFDE on with initial data , where

Assume furthermore that and , so system (5) has the solution . The solution is called the trial solution or equilibrium solution.

To get our main results, we firstly put forward the following hypothesis.

*Assumption 1 (the local Lipschitz condition). *For each integer , there exists a positive constant such that
for all , with .

*Remark 2. *By Theorem 3.1 in [15] or Lemma 2.3 in [16], this assumption with conditions and can guarantee a unique maximal local solution to system (5) for any initial data.

However, to ensure the unique maximal local solution is in fact the global solution, we need to impose the following additional polynomial growth condition.

*Assumption 3 (the polynomial growth condition). *There exist constants , probability measures on , , and positive numbers , satisfying , and bounded functions such that
for all , .

*Remark 4. *The probability measures , , can be weakened to any right continuous nondecreasing functions (see [19]). Compared with [10, 11], Assumption 3 in this paper is a generalization of Assumption 3 of [10] and Assumption 2 of [11].

Let denote the family of all continuous nonnegative functions on , which are continuously twice differentiable in and once differentiable in . For each , denote an operator from to by where , , and //.

Then let us recall a number of lemmas.

Lemma 5 (cf. [20]). *If is a bounded function on and , then for any , .*

Lemma 6 (cf. [11]). *Assume , , . If the following condition holds,
**
then there exists satisfying
**
for all .*

Lemma 7 (cf. [14]). *Assume . For any , if , then there exists a constant satisfying
*

Lemma 8 (Kolmogorov-Chentsov theorem [21]). *Suppose that a stochastic process on satisfies the condition
**
for some positive constants , , and . Then there exists a continuous modification of , which has the property that, for every , there is a positive random variable such that
**
In other words, almost every sample path of is locally but uniformly Hlder-continuous with exponent .*

#### 3. Boundedness of SFDEs

For a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of this equation are generally required to satisfy the linear growth condition and the local Lipschitz condition (see [2, 4, 5]) or a given non-Lipschitz condition and the linear growth condition (see [22]). However, when the coefficients of the system (5) satisfy the local Lipschitz condition and the polynomial growth condition, the solution of the system (5) may explode at a finite time. So it is necessary to examine the existence and uniqueness of the global solution of the system (5). Here we state the following existence-and-uniqueness result.

Lemma 9. *If Assumptions 1 and 3 and hold, then for any initial data , there is a unique global solution of system (5) on .*

*Remark 10. *This result is the special case of Theorem 3.2 of [15]. Since it is not so easy to see this fact directly, we give the proof in the Appendix. The fact that we write down our Lemma 9 here is to keep our paper completely based on Assumptions 1 and 3.

We now show the following asymptotic boundedness of the global solution in the sense of the pth moment and the trajectory with large probability.

Theorem 11. *If Assumptions 1 and 3 and hold, then for any initial data and any , the global solution of system (5) is bounded in the sense of th moment; that is, there exists a constant such that
*

*Proof. *Since , the existence and uniqueness of the solution follow from Lemma 9. And there exists at least a sufficiently small positive constant satisfying . So by the continuity, define . For the sake of simplicity, write , . For any , applying Itô’s formula to , we yield
where , , , and .

Noting that and for any , by Lemma 7 and the same technique as (A.3), , as a function of , has a positive upper-boundedness; that is, there is a positive constant such that

And in view of the fact that
for , , and , we yield that, for ,
By virtue of the boundedness of , there is a constant such that , which implies that
where / + + . This implies
From the boundedness of initial data , we claim that, for any , there exists a constant such that . When , using the Lyapunov inequality, we claim that

From Theorem 11 and the Chebyshev inequality, we get the following proposition about the asymptotic boundedness of the global solution in the sense of the trajectory with large probability.

Proposition 12. *If Assumptions 1 and 3 and hold, then for any initial data , the global solution of system (5) is stochastically ultimately bounded; namely, for any , there exists a constant such that
*

*Proof. *For any , letting , applying Theorem 11 and the Chebyshev inequality, we have
So we get the required assertion.

Further we continue to discuss the asymptotic boundedness of the norm of in system (5) in the sense of the th moment and the trajectory with large probability.

Theorem 13. *If Assumptions 1 and 3 and hold, then for any initial data , the norm of in system (5) is bounded in the sense of th moment; that is, there exists a constant such that the global solution of system (5) has the property
*

*Proof. *For the sake of simplicity, write . From Theorem 11, set . For , and , using Itô’s formula, we compute that
where . Using Young’s inequality, we compute that
where , , + , and . However we can get , for , and , respectively.

Let + − . Since , by the same technique as (A.3) in the Appendix, we get that there exists a positive constant such that . By virtue of the boundedness of , assuming , we have

By virtue of the boundedness of , we have

Therefore, from (26), we have

Using B-D-G inequality, we get that
where the boundedness of can be obtained from the boundedness of and above and is a constant which is not necessary to know exactly. Substituting it into (30), we yield

So we claim that there exists a such that for any . When , using the Lyapunov inequality, we claim that

*Remark 14. *Clearly, the key of the proof is the upper-boundedness of function , which depends on the condition . And the theorem will play an important role to ensure the almost sure asymptotic stability of the solution.

In the same way as Proposition 12, we get the following proposition.

Proposition 15. *If Assumptions 1 and 3 and hold, then for any initial data , the norm of in system (5) is stochastically ultimately bounded; namely, for any , there exists a constant such that the global solution of system (5) has the property
*

#### 4. Almost Sure Asymptotic Stability of SFDEs

In this section, we aim to study the almost sure asymptotic stability of system (5). The following theorem establishes new criteria on the almost sure asymptotic stability.

Theorem 16. *If Assumptions 1 and 3 and the following condition (35) hold,
**
then for any initial data , there is a unique global solution of system (5) on , and is almost surely asymptotically stable; that is,
**
where .*

*Proof. *The existence and uniqueness of the global solution follow from Lemma 9 directly. For the sake of simplicity, write . Applying Itô’s formula to , we yield
where , , , , and , and we have used the elemental inequality: for any , ,
Let . Since /, there exists such that . Setting , , satisfying , / > . By using Lemma 6, we get that there exists a constant satisfying . Then choose which is sufficiently close to 1 such that
We therefore have
In view of the fact that , for , , and , respectively, we get
where , which is a local martingale with the initial value . From Lemma 5, we get . Applying the nonnegative semimartingale convergence theorem (see [23]), we obtain that

To obtain our main result, we need to claim that almost every sample path of is uniformly continuous on . Let , where , . From (42), we get that there is a constant such that . By the boundedness of and initial data , we get that there is a constant such that . By virtue of the boundedness of , , assume . From Assumption 3, it is easy to conclude that
Then we get
for any . This means is uniformly continuous on .

For any , using B-D-G inequality, from Theorems 11 and 13, we get
From the Kolmogorov-Chentsov theorem (see Lemma 8), we obtain that almost every sample path of is locally but uniformly Hlder-continuous with exponent for every . So we have that almost every sample path of is uniformly continuous. Therefore, we claim that almost every sample path of is uniformly continuous on . Then from Barbalat Lemma (see [20]) and (42), we claim that

*Remark 17. *Clearly, the key of the proof is the positive lower-boundedness of function , which depends on condition (35). Since the positive lower-boundedness of can guarantee (41), so we can use the nonnegative semimartingale convergence theorem to get the asymptotic stability.

*Remark 18. *From the proof above, Assumptions 1 and 3 are enough to guarantee the asymptotic stability of system (5). And the coefficients of system (2) do not satisfy the conditions which are similar to Assumptions 2 of [10] or Assumptions 3 of [11]. So compared with [10, 11], the three conditions of guaranteeing the asymptotic stability are weakened to the two conditions by this paper.

#### 5. Example

In this section, we will discuss some examples to illustrate our results.

*Example 1. *Let us return to the SFDE (2). We can compute that
So we obtain that , and condition (35) holds. Through Theorems 13 and 16, the global solution of system (2) has the following properties:
where is some positive constant.

*Example 2. *Let us consider the scalar SFDE as follows:
with initial data , where is a scalar Brownian motion. And are bounded linear operators from to satisfying , where are probability measures on , .

We compute that