Stability of Nonlinear Neutral Stochastic Functional Differential Equations

Volume 2010, Article ID 425762,26pages doi:10.1155/2010/425762

Research Article

Stability of Nonlinear Neutral Stochastic Functional Differential Equations

Minggao Xue,

Shaobo Zhou,

and Shigeng Hu

1School of Management, Huazhong University of Science and Technology, Wuhan 430074, China

2School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Correspondence should be addressed to Shaobo Zhou,[email protected] Received 17 June 2010; Revised 14 September 2010; Accepted 18 September 2010 Academic Editor: Neville Ford

Copyrightq2010 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.

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:

dxt−uxt fxt, tdtgxt, tdwt. 1.1

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 see1, page 202, Theorem 2.2.

Theorem 1.1. Assume that there exist positive constantsK,L,andκ∈0,1such that

f ϕ, t

−f

ψ, t²∨g ϕ, t

−g

ψ, t²≤Kϕ−ϕ², f

ϕ, t²∨g

ϕ, t² ≤L

1ϕ² , u

ϕ

−u

ψ≤κϕ−ψ

1.2

for allϕ, ψ ∈ C−τ,0;Rⁿ.Then there exists a unique solutionxtto1.1with initial dataξ ∈ C^b_F

0−τ,0;Rⁿ(i.e.,ξis anF0-measurableC−τ,0;Rⁿ-valued random variable such thatEξ<

∞).

Theorem 1.1requires that the coefficientsfandgsatisfy 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:

dxt−uxt xt

abσ1xt−xt² dtcxtσ2xtdwt, σ₁

ϕ∨σ₂ ϕ< κ

₀

−τ

ϕθdμθ,

1.3

where coefficients fx, xt, t xtabσ₁xt−xt² and g cxtσ2xt 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 that1.3has 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. Letc₁, c₂, λ, pbe all positive numbers andq > c2/c₁1−κ^−p, κ ∈ 0,1,for any ϕ∈L^p_Ft

0−τ,0;Rⁿ,

Eu

ϕ^p≤κ^pϕ^p

0, 1.4

and assume that there exists a functionVx, t∈C^2,1Rⁿ×−τ,∞;Rsuch that

c₁|x|^p≤Vx, t≤c₂|x|^p 1.5

for allx, t∈Rⁿ×−τ,∞and also for allt≥0 ELV

ϕ, t

≤ −λEV ϕ0, t

1.6

providedϕ{ϕθ:−τ ≤θ≤0} ∈L^p_Ft−τ,0;Rⁿsatisfying EV

ϕθ, tθ

≤qEV ϕ0, t

1.7

for all−τ ≤θ≤0.Then for allξ∈C_F^b

0−τ,0;Rⁿ, t≥0

E|xt;ξ|^p≤ c₂ c1

1κ 1−κ1

_p

e^−γtEξ^p₀, κ₁κe^γτ/p, 1.8

whereγμ∧τ⁻¹lnq11κq₁^1/p−p, q1c1q/c2.

It is very difficult to verify the conditions of Theorem 1.2, and it is clear that ELVϕ, t≤ −λEVϕ0, t does not hold for many NSFDEs. In fact, for1.3, if one chooses Vx, t x²,then

LV 2xx−uxt

abσ₁xt−x²

c²x²σ₂²xt. 1.9

Here, the polynomial x⁴ appears on the right-hand side, and it has an order of 4 which is higher than the order ofVx x².More recently, Mao3–5, Zhou et al.6,7, Yue et al.8 and Shen et al.9provided 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 operatorLV 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 10and Mao and Rassias11results 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 5gives 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 Ω,F,{Ft}_t≥0, P be a complete probability space with a filtration {Ft}_t≥0, satisfying the usual conditions i.e., it is right continuous and F0 contains all P-null sets. Let wt w1t, . . . , wmt^T be an m- dimensional continuous local martingale with w0 0 defined on the probability space.

IfAis a vector or matrix, its transpose is denoted byA^T. IfAis a matrix, its trace norm is denoted by |A|

traceA^TA, while its operator norm is denoted byA sup{|Ax| :

|x|1}without any confusion withϕ.C−τ,0;Rⁿdenote the family of all continuous functions ϕ from −τ,0 to Rⁿ with the norm ϕ sup_{−τ≤θ≤0}|ϕθ|, where | · | is the Euclidean norm inRⁿ.Denoted byC_F^b

0−τ,0;Rⁿthe family of all bounded,F0-measurable, C−τ,0;Rⁿ-valued random variables.

Consider an n-dimensional neutral stochastic functional differential equation

dxt−uxt, t fxt, tdtgxt, tdwt 2.1

ont≥0 with initial datax₀ξ∈C^b_F

0−τ,0;Rⁿ, u:C−τ,0;Rⁿ → Rⁿ,and

f :C−τ,0;Rⁿ×R−→Rⁿ, g :C−τ,0;Rⁿ×R −→R^n×m 2.2

are Borel measurable. Letxt;ξdenote the solution of2.1whilex_t{xtθ:−τ≤θ≤0}

which is regarded as aC−τ,0;Rⁿ-valued stochastic process, denoted byxt xt−uxt. Let C^2,1Rⁿ ×R;Rdenote the family of all nonnegative functions Vx, t onRⁿ × R which are continuously twice differentiable inxand once differentiable int. IfVx, t ∈ C^2,1Rⁿ×R;R,define an operatorLV :C−τ,0;Rⁿ×RtoRby

LV ϕ, t

Vt ϕ0, t

Vx ϕ0, t

f ϕ, t

1 2trace

g^T ϕ, t

Vxx ϕ0, t

g ϕ, t

, 2.3

where V_tx, t ∂Vx, t/∂t,V_xx, t ∂Vx, t/∂x1,∂Vx, t/∂x2, . . . , ∂Vx, t/∂xn, V_xxx, t∂²Vx, t/∂xix_jn×n.

For the purpose of stability, assume thatf0, t g0, t u0, t 0.This implies that 2.1 admits a trivial solution, x0, t 0. Furthermore, we impose the following assumptions.

H1 The local Lipschitz condition. For each integerR ≥ 1,there is a positive constantK_Rsuch that

f ϕ, t

−f

ψ, t²∨g ϕ, t

−g

ψ, t²≤KRϕ−ψ² 2.4 for thoseϕ, ψ∈C−τ,0;Rⁿwithϕ ∨ ψ ≤Randt∈R.

H2There exists a positive constantκ ∈ 0,1and a probability measure νsuch that

u ϕ, t

−u

ψ, t≤κ ₀

−τ

ϕ−ψdνθ 2.5 for anyϕ, ψ∈C−τ,0;Rⁿ.

H3 There are two functions V ∈ C^2,1Rⁿ × −τ,∞;R and U ∈ CRⁿ ×

−τ,∞;Ras well as positive constantsλ₁, λ₂, c₁, c₂and a probability measureμ on−τ,0such that

c₁|x|²≤Vx, t≤c₂|x|², 2.6

LV ϕ, t

≤λ1

1V

ϕ0, t

₀

−τ

V

ϕθ, tθ U

ϕθ, tθ dμθ

−λ2U ϕ0, t

2.7

for all−τ ≤θ≤0, ϕ, t∈C−τ,0;Rⁿ×R.

Remark 2.1. In condition2.7, we see that the functionUx, tplays a key role in allowing coefficientsfandgto be nonlinear functions.

Theorem 2.2. Assume that (H1), (H2), and (H3) hold. Then for any initial condition ξ ∈ C^b_F

0−τ,0;Rⁿ,there exists a unique global solution xtto2.1ont ∈ −τ,∞.Moreover, the solution has the properties that

EVxt, t<∞, E _t

0

Uxs, sds <∞ 2.8

for anyt≥0.

Proof. It is clear that for any initial dataξ ∈ C^b_F

0−τ,0;Rⁿ,there exists a unique maximal local solutionxtont∈−τ, τe,whereτ_eis the explosion time1, by applying the standing truncation techniquesee Mao12,13to2.1. According toH2, we have

|x0| ≤ |x0||ux₀,0| ≤ |x0|κ ₀

−τ|xθ|dνθ≤1κξ. 2.9

Letk₀>1κξbe sufficiently large such that 1

k0

< min

−τ≤t≤0|xt|< max

−τ≤t≤0|xt| < k₀. 2.10

Define the stopping time τkinf

t∈0, τe:|xt| ∈ Ik , Ik≡

1 k, k

, k≥k₀, 2.11

where throughout this paper, we set inf∅ ∞∅ denotes the empty sets. Clearly, τk is increasing ask → ∞.Denoteτ_∞ limk→ ∞τk, τ_∞ ≤ τe a.s. We will show thatτe ∞a.s., which implies thatxtis global.

It ˆo formula and condition2.7yield

dVxt, t LVx, tdtV_xx, tg xt, tdwt

≤λ₁

1Vxt, t ₀

−τ

Vx_t, tθdμθ ₀

−τ

Ux_t, tθdμθ

dt

−λ2Uxt, tdtVxx, tgx t, tdwt

2.12

fort≥ 0.For anyk≥ k₀andt∈0, τ,we integrate both sides of2.12from 0 toτk∧tand then take the expectations to get

EVxτk∧t, τk∧t−Vx0, 0

≤E _τk∧t

0

λ₁

1 ₀

−τVxs, sθ Uxs, sθdμθ

ds

E _τk∧t

0

λ1Vxs, s−λ₂Uxs, sds.

2.13

According to the integral substitution technique, we estimate

_τk∧t

0

₀

−τVxs, sθdμθds

_τk∧t

0

₀

−τVxsθ, sθdμθds

≤ ₀

−τdμθ _τk∧tθ

θ Vxs, sds≤ _τk∧t

−τ Vxs, sds

≤ _τ

−τVxs, sds <∞,

2.14

Similiarly,

_τk∧t

0

₀

−τUxs, sθdμθds≤ _τ

−τUxs, sds <∞. 2.15

Substituting for2.14and2.15into2.13, and by using the Fubini theorem, the result is

EVxτ k∧t, τk∧t≤Vx0, 0 λ₁τλ₁E _τk∧t

0

Vxs, sds

−λ₂E _τk∧t

0

Uxs, sdsλ₁E _τ

−τVxs, s Uxs, sds

≤C₁λ₁E _τk∧t

0

Vxs, sds

≤C₁λ_{1 t}

0

EVxτ_k∧s, τ_k∧sds,

2.16

whereC₁ Vx0, 0 λ₁τλ₁E_τ

−τVxs, s Uxs, sds.Equations 2.6and2.9 implyVx0, 0≤c₂1κ²ξ²; thus,C₁ is a finite constant. By using inequalityab² ≤ 1/1−κ₀a² 1/κ0b², a, b >0, κ₀∈0,1; thus,

E|xτk∧t|²≤1−κ₀⁻¹E|xτ k∧t|²κ⁻¹₀ E|uxτk∧t, τk∧t|², 2.17

condition2.6yields

c⁻¹₂ Vxτ k∧t, τk∧t≤ |xτ k∧t|² ≤c⁻¹₁ Vxτ k∧t, τk∧t. 2.18

H2and the H ¨older inequality yield

E|uxτk∧t, τk∧t|² ≤κ²E ₀

−τ

ϕθdνθ 2

≤κ² ₀

−τE|xτk∧tθ|²dνθ. 2.19

Substituting for2.16,2.18, and2.19into2.17, the result is

E|xτk∧t|²≤1−κ0⁻¹c₁⁻¹

C1λ1

_t

0

EVxτk∧s, τk∧sds

κ⁻¹₀ κ² ₀

−τE|xτk∧tθ|²dνθ.

2.20

For anyt∈−τ, τ,2.20implies

−τ≤s≤tsupE|xτk∧s|²≤1−κ0⁻¹c⁻¹₁

C1λ1

_t

0

EVxτk∧s, τk∧sds

κ⁻¹₀ κ² sup

−τ≤s≤tE|xτk∧s|². 2.21

Letκ0κ,then

sup

−τ≤s≤tE|xτk∧s|²≤1−κ⁻²c⁻¹₁

C₁λ_{1 t}

0

EVxτk∧s, τk∧sds

. 2.22

Therefore, for anyt∈−τ, τ,

E|xτ_k∧t|²≤1−κ⁻²c₁⁻¹

C₁λ_{1 t}

0

EVxτ_k∧s, τ_k∧sds

. 2.23

By2.6, we may obtain

EVxτk∧t, τk∧t≤c2E|xτk∧t|²≤1−κ⁻²c⁻¹₁ c2

C1λ1

_t

0

EVxτk∧s, τk∧sds

. 2.24

For anyk≥k₀, t∈0, τ,the Gronwall inequality implies

EVxτk∧τ, τk∧τ≤c₁⁻¹c₂C₁1−κ⁻²e^{c−11c2λ11−κ−2t}. 2.25

Thus, for allk≥k₀,

EVxτk∧τ, τk∧τ≤c⁻¹₁ c₂C₁1−κ⁻²e^{c1−1c2λ11−κ−2τ}, 2.26

which implies

EVxt, t≤c₁⁻¹c₂C₁1−κ⁻²e^{c−11c2λ11−κ−2τ}, 0≤t≤τ. 2.27

SinceEI{τk≤τ}Vxτk∧τ, τk∧τ≤EVxτk∧τ, τk∧τ, and definingμkinf|x|≥k,0≤t<∞Vx, t fork≥k₀,according to2.26, then

μkPτk≤τ≤c⁻¹₁ c₂C₁1−κ⁻²e^{c−11c2λ11−κ−2τ}. 2.28

Clearly, condition2.6implies lim_{k→ ∞}μ_k ∞.Lettingk → ∞in2.28, thenPτ_∞≤τ 0, namely,

Pτ_∞> τ 1. 2.29

Moreover, settingtτin2.16, we may obtain that λ₂E

_τk∧τ

0

Uxs, sds≤C₁λ₁E _τk∧τ

0

Vxs, sds≤C₁λ₁τc⁻¹₁ c₂C₁1−κ⁻²e^{c−11c2λ11−κ−2τ}, 2.30

that is,

E _τ

0

Uxs, sds≤ C₁ λ₂

1λ₁τc₁⁻¹c₂C₁1−κ⁻²e^{c−11c2λ11−κ−2τ}

<∞. 2.31

Let us now proceed to proveτ_∞ > 2τ a.s. given that we have shown2.27–2.31. For any k ≥ k0 andt ∈ 0,2τ,we can integrate both sides of2.12from 0 to τk∧tand then take expectations to get

EVxτ k∧t, τk∧t≤C₂λ₁E _τk∧t

0

Vxs, sds−λ₂E _τk∧t

0

Uxs, sds, 2.32

where

C2Vx0 2λ1τλ1E _2τ

−τVxs, s Uxs, sds <∞. 2.33 By the Gronwall inequality and2.32, we have

EVxτ k∧t, τk∧t≤c⁻¹₁ c₂C₂1−κ⁻²e^{c−11c2λ11−κ−2t}, 0≤t≤2τ, k≥k₀. 2.34

In particular,

EVxτ k∧2τ, τk∧2τ≤c⁻¹₁ c₂C₂1−κ⁻²e^{c1−1c2λ11−κ−22τ}, ∀k≥k₀. 2.35

This implies

μkPτk≤2τ≤c⁻¹₁ c2C21−κ⁻²e^{c−11c2λ11−κ−22τ}. 2.36

Lettingk → ∞, by2.6, thenPτ∞≤2τ 0,that is,

Pτ_∞>2τ 1, EVxt, t≤c⁻¹₁ c₂C₂1−κ⁻²e^{c−11c2λ11−κ−22τ}, 0≤t≤2τ. 2.37

By2.32, we may obtain that

λ₂E _τk∧2τ

0

Ux, tdt≤C₂λ₁E _τk∧2τ

0

Vxt, tdt, 2.38

that is,

E _2τ

0

Ux, tdt≤ C₂ λ2

12λ₁τc⁻¹₁ c₂1−κ⁻²e^{c−11c2λ11−κ−22τ}

<∞. 2.39

Repeating this procedure, we can show that, for any integeri≥1, τ_∞> iτa.s. andEVxt, t≤ c⁻¹₁ c2Ci1−κ⁻²e^{c−11c21−κ−2λ1iτ},0≤t < iτ,and

E _iτ

0

Ux, tdt≤ Ci

λ₂

1iλ1τc⁻¹₁ c21−κ⁻²e^{c−11c21−κ−2λ1iτ}

, 2.40

where

CiVx0, 0 λ1E _iτ

−τ1Vx, t Ux, tdt <∞. 2.41 We must therefore haveτ_∞∞a.s. as well as the required assertion.

Note that condition2.6may be replaced by more general conditionc₁|x|^p≤Vx, t≤ c2|x|^p, p≥2,which is suitable to the corresponding results below.

3. Boundedness and Moment Stability

In the previous section, we have shown that the solution of2.1has the properties that

EVxt, t<∞, E _t

0

Uxs, sds <∞ 3.1

for any t ≥ 0. 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 to2.1under specified conditions.

Theorem 3.1. Assume that (H1), (H2), and (H3) hold except2.7which is replaced by

LV ϕ, t

≤μ₁−μ₂V ϕ0, t

μ_{3 0}

−τV

ϕ, tθ

dη₁θ−μ₄U ϕ0, t

μ5

₀

−τU

ϕ, tθ

dη2θ−μ6V

ϕ0, t 3.2

for allϕ, t ∈C−τ,0;Rⁿ×R, −τ ≤θ ≤ 0,whereμ₁ ≥ 0, μ₂ > μ₃ ≥0, μ₄ > μ₅ >0, μ₆ >0 are constants andη₁θandη₂θare probability measures on−τ,0.Then for any initial dataξ, the global solutionxtto2.1has the property that

lim sup

t→ ∞EVxt, t< c₂μ₁

1−κ₀²c₁ε, 3.3

whereεμ₆∧ε₁∧ε₂∧τ⁻¹logκ⁻², κ₀ κ√

e^ετ,whileε₁>0 andε₂ >0 are the unique roots to the following equations:

μ2μ3e^ε1τ, μ4μ5e^ε2τ, 3.4

respectively. Ifμ10,then

lim sup

t→ ∞

1

t lnEVxt, t<−ε,

_∞

0

EUxt, tdt <∞. 3.5

Proof. We first observe that3.2implies2.7if we setλ1 μ1∨μ3∨μ5andλ2 μ4.So, for any initial data,2.1has a unique global solutionxtont ≥ −τ,which has the properties 2.8. Based on these properties, we can apply the It ˆo formula and condition3.2to obtain that for anyt≥0,

d

e^εtVxt, t

e^εtεVxt, t LVxt, tdte^εtVxxt, tgxt, tdwt

≤e^εt

μ1−μ2Vx, t μ3

₀

−τVxt, tθdη1θ−μ4Ux, t μ5

₀

−τUxt, tθdη2θ

e^εtVxxt, tg xt, tdwt−e^εt μ₆−ε

Vxt, t,

3.6

We integrate both sides of the above inequality from 0 totand take expectations to get

e^εtEVxt, t

≤Vx0, 0 μ₁e^εt ε −μ₂E

_t

0

e^εsVxs, sdsμ₃E _t

0

₀

−τe^εsVxs, sθdη1θds

−μ4E _t

0

e^εsUxs, sdsμ5E _t

0

₀

−τe^εsUxs, sθdη2θds,

3.7

by using ofεμ₆∧ε₁∧ε₂< μ₆.Compute

E _t

0

₀

−τe^εsVxs, sθdη1θdsE _t

0

₀

−τe^εsVxsθ, sθdη1θds

≤e^ετE _t

0

₀

−τe^εsθVxsθ, sθdη1sθds

≤e^ετE ₀

−τdη1θ _tθ

θ e^εsVxs, sdds

≤e^ετE _t

−τe^εsVxs, sds

≤e^ετE ₀

−τe^εsVxs, sdse^ετE _t

0

e^εsVxs, sds, 3.8

Similiarly,

E _t

0

₀

−τe^εsUxs, sθdη2θds≤e^ετE ₀

−τe^εsUxs, sdse^ετE _t

0

e^εsUxs, sds. 3.9

Substituting for3.8and3.9into3.7, the result is

e^εtEVxt, t

≤Vx0, 0 μ₁e^εt ε −μ2E

_t

0

e^εsVxs, sds

μ₃e^ετE ₀

−τe^εsVxs, sdsμ₃e^ετE _t

0

e^εsVxs, sds

−μ4E _t

0

e^εsUx, sdsμ5e^ετE ₀

−τe^εsUxs, sdsμ5e^ετE _t

0

e^εsUxs, sds

Vx0, 0 μ₁e^εt ε e^ετE

₀

−τe^εs

μ₃Vxs, s μ₅Uxs, s ds

−

μ₂−μ₃e^ετ E

_t

0

e^εsVxs, sds−

μ₄−μ₅e^ετ E

_t

0

e^εsUx, sds

Cμ₁e^εt ε −

μ₂−μ₃e^ετ E

_t

0

e^εsVxs, sds−

μ₄−μ₅e^ετ E

_t

0

e^εsUx, sds,

3.10

whereCVx0, 0e^ετE₀

−τe^εsμ3Vxs, sμ₅Uxs, sds.It is clear that, forε≤ε₁∧ε2, we haveμ₂−μ₃e^ετ ≥0, μ₄−μ₅e^ετ≥0,hence,

e^εtEVxt, t ≤Cμ1e^εt

ε . 3.11

ByH2andH3and inequalityab² ≤1/1−κ0a² 1/κ0b², a, b >0, κ0∈0,1,we compute

Ee^εt|xt|²≤1−κ₀⁻¹Ee^εt|xt| ²κ⁻¹₀ Ee^εt|uxt, t|²

≤1−κ₀⁻¹c⁻¹₁ e^εtE|Vxt, t|²κ⁻¹₀ κ² ₀

−τe^εtE|xtθ|²dνθ

≤1−κ₀⁻¹c⁻¹₁

Cμ₁ ε e^εt

κ⁻¹₀ κ² ₀

−τe^−εθEe^εtθ|xtθ|²dνθ.

3.12

For anyt≥0,

sup

−τ≤s≤tEe^εt|xt|² ≤1−κ₀⁻¹c⁻¹₁

Cμ₁ ε e^εt

κ⁻¹₀ κ²sup

−τ≤s≤tEe^εt|xt|² ₀

−τe^−εθdνθ

≤1−κ0⁻¹c⁻¹₁

Cμ₁ ε e^εt

κ⁻¹₀ κ²e^ετ sup

−τ≤s≤tEe^εt|xt|².

3.13

Letingκ₀κ√

e^ετ <1,sinceε < τ⁻¹logκ⁻²,thenκ₀ <1,

−τ≤s≤tsupEe^εt|xt|²≤1−κ₀⁻²c₁⁻¹

Cμ₁ εe^εt

, 3.14

and byH3,EVx, t≤c2E|xt|²,then

sup

−τ≤s≤te^εtEVx, t≤c₂sup

−τ≤s≤te^εtE|xt|²≤1−κ₀⁻²c⁻¹₁ c₂

Cμ₁ ε e^εt

. 3.15

Therefore,

lim sup

t→ ∞EVxt, t≤ c₂μ₁

1−κ₀²c₁ε. 3.16

Whenμ₁0,thenEVxt, t≤c⁻¹₁ c₂C1−κ₀⁻²e^−εt,for all t≥0,that is,

lim sup

t→ ∞

1

tlogEVxt, t≤ −ε. 3.17

On the other hand, whenμ₁0,by3.7and the It ˆo formula, we may show easily that

EVxt, t Vx0, 0 E ₀

−τ

μ₃Vxs, s μ₅Uxs, s ds

−

μ₂−μ₃ E

_t

0

Vxs, sds−

μ₄−μ₅ E

_t

0

Uxs, sds.

3.18

Byμ₂μ₃e^ε1τ > μ₃, μ₄μ₅e^ε2τ > μ₅,and the Fubini theorem, we obtain _t

0

EUx, s≤ 1 μ₄−μ₅

Vx0,0 E ₀

−τ

μ₃Vxs, s μ₅Uxs, s ds

<∞. 3.19

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 con- ditions to guarantee asymptotic stability. Therefore we introduce the following semitingale convergence thoerem14,15, which will play a key role in dealing with nonlinear systems.

Lemma 4.1. LetMtbe a real-valued local martingale withM0 0 a.s. Letζbe a nonnegative F0-measurable random variable. IfXtis a nonnegative continuousFt-adapted process and satisfies Xt≤ζMtfort≥0,thenXtis almost surely bounded, namely, lim_{t→ ∞}Xt<∞,a.s.

Theorem 4.2. Assume that (H1), (H2), and (H3) hold except2.7which is replaced by

LV ϕ, t

≤ −μ2V ϕ0, t

μ3

₀

−τV

ϕ, tθ

dη1θ−μ4U ϕ0, t

μ_{5 0}

−τU

ϕ, tθ

dη₂θ−μ₆V

ϕ0, t 4.1

for allϕ, t∈Rⁿ×R,−τ ≤θ ≤0,whereμ₂ > μ₃ ≥ 0, μ₄ > μ₅ > 0, μ₆ >0.Then, for any initial data, the unique global solutionxtof2.1has the property that

lim sup

t→ ∞

1

t lnVxt, t≤ −ε,

_∞

0

Uxt, tdt <∞, 4.2

whereε μ₆∧ε₁∧ε₂∧τ⁻¹logκ⁻²,whileε₁ >0 andε₂ >0 are the unique roots to the following equations:

μ2μ3e^ε1τ, μ4μ5e^ε2τ, 4.3

respectively.

Proof. We first observe that4.1implies2.7if we setλ1 μ1∨μ3∨μ5andλ2 μ4.So, for any initial data,2.1has a unique global solutionxtont ≥ −τ,which has the properties 2.8. Similar to the proof ofTheorem 3.1, applying the It ˆo formula and condition4.1, for anyt≥0,we may obtain that

d

e^εtVxt, t

e^εtεVxt, t LVxt, tdte^εtV_xxt, tgx t, tdwt

≤e^εt

−μ2Vxt, t μ_{3 0}

−τ

Vx_t, tθdη1θ−μ₄Ux, t μ_{5 0}

−τ

Ux_t, tθdη2θ

e^εtVxxt, tgxt, tdwt− μ6−ε

Vxt, t.

4.4

Fort >0,we can integrate both sides of the above inequality from 0 totand take expectations to get

e^εtEVxt, t≤Vx0, 0−μ_{2 t}

0

e^εsVxs, sdsμ_{3 t}

0

₀

−τe^εsVxs, sθdη1θds

−μ4

_t

0

e^εsUx, sdsμ5

_t

0

₀

−τe^εsUxs, sθdη2θdsMt,

4.5

whereMt _t

0e^εsVxxs, sgx s, sdwsdsis a real-valued continuous local martingale withM0 0.Similar toTheorem 3.1, we have

e^εtVxt, t ≤Vx0,0 e^ετ ₀

−τ

μ₃Vxs, s μ₅Uxs, s ds

−

μ₂−μ₃e^ετ

t 0

e^εsVxs, sds−

μ₄−μ₅e^ετ

t 0

e^εsUx, sdsMt

≤constMt.

4.6

Lemma 4.1implies

lim sup

t→ ∞e^εtVxt, t<∞ a.s. 4.7

Sincec₁|x|²≤Vx, t≤c₂|x|²,then

lim sup

t→ ∞e^εt|xt|² <∞ a.s. 4.8

According to the definition ofxt, we compute

e^εt|xt|²e^εt|xt uxt, t|²

≤1−κ₀⁻¹e^εt|xt| ²κ⁻¹₀ e^εt|uxt, t|²

≤1−κ₀⁻¹e^εt|xt| ²κ⁻¹₀ κ² ₀

−τe^−εθe^εtθ|xtθ|²dνθ.

4.9

Therefore, we may also compute

−τ≤s≤tsupe^εs|xs|²≤ ξ²sup

0≤s≤te^εs|xs|²

≤ ξ² 1−κ0⁻¹sup

0≤s≤t

e^εs|xs| ²κ⁻¹₀ κ² ₀

−τe^−εθe^εsθ|xsθ|²dνθ

≤ ξ² 1−κ0⁻¹sup

0≤s≤te^εs|xs| ²κ⁻¹₀ κ² sup

−τ≤s≤te^εs|xs|² ₀

−τe^−εθdνθ

≤ ξ² 1−κ₀⁻¹sup

0≤s≤te^εs|xs| ²κ⁻¹₀ κ²e^ετ sup

−τ≤s≤te^εs|xs|².

4.10

Noting thatε < τ⁻¹logκ⁻²,chooseκ₀κ√

e^ετ.Thenκ₀<1,and we obtain

sup

−τ≤s≤te^εs|xs|²≤1−κ₀⁻¹ξ² 1−κ₀⁻²sup

0≤s≤te^εs|xs|². 4.11

4.8and4.11yield

sup

−τ≤s≤te^εs|xs|²<∞ a.s. 4.12

Recall the conditionc₁|x|² ≤ Vx, t≤ c₂|x|²,which implies lim sup_{t→ ∞}e^εtVxt, t < Ca.s.

The required result is obtained.

Remark 4.3. From the processes of the proof of Theorems3.1and4.2, we see that condition 2.6plays an important role in dealing with the neutral term. Moreover, applying condition 2.6, we can also obtain more precise results

lim sup

t→ ∞

1

t logE|xt| ≤ −ε

2, lim sup

t→ ∞

1

t log|xt| ≤ −ε

2. 4.13

In the next section, condition2.6will be replaced by a more general condition for stochastic functional differential equation.

5. Stochastic Functional Differential Equation

Letuxt 0. Then2.1reduces to

dxt fxt, tdtgxt, tdwt. 5.1

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 operatorLV fromC−τ,0;Rⁿ×RtoRby

LV ϕ, t

V_t ϕ0, t

V_x ϕ0, t

f ϕ, t

1 2trace

g^T ϕ, t

V_xx ϕ0, t

g ϕ, t

. 5.2

We impose the following assumption which is more general thanH3.

H3 There are two functions V ∈ C^2,1Rⁿ × −τ,∞;R and U ∈ CRⁿ ×

−τ,∞;R as well as two positive constantsλ1, λ2 and a probability measure μon−τ,0such that

|x| → ∞lim inf

0≤t<∞Vx, t ∞, 5.3

LV ϕ, t

≤λ1

1V

ϕ0, t

₀

−τ

V

ϕθ, tθ U

ϕθ, tθ dμθ

−λ2U ϕ0, t

5.4

for all−τ ≤θ≤0,ϕ, t∈Rⁿ×R.

Theorem 5.1. Assume that (H1) and (H3) hold. Then for any initial conditionξ∈C^b_F

0−τ,0;Rⁿ, there exists a unique global solution xtof 5.1on t ∈ −τ,∞.Moreover, the solution has the properties that

EVxt, t<∞, E _t

0

Uxs, sds <∞ 5.5

for anyt≥0.

Proof. Since the proof is similar to Theorem 2.2, we will only outline the proof. It is clear that for any initial dataξ∈C^b_F

0−τ,0;Rⁿ,there is a unique maximal local solutionxton t∈−τ, τe,whereτeis the explosion time1. Letk₀>0 be sufficiently large for

1 k0

< min

−τ≤t≤0|xt|< max

−τ≤t≤0|xt|< k₀. 5.6

Define the stopping time

τ_kinf

t∈0, τe:|xt| ∈ I_k , I_k≡

1 k, k

, k≥k₀, 5.7