# Stability and Boundedness of Stochastic Volterra Integrodifferential Equations with Infinite Delay

Authors: Chunmei Zhang; Wenxue Li; Ke Wang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We make the first attempt to discuss stability and boundedness of solutions to stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). By the Lyapunov-Krasovskii functional approach, we get kinds of sufficient criteria for stability and boundedness of solutions to IDSVIDEs. The main innovation here is that stochastic systems with infinite delay can retain stability and boundedness of corresponding deterministic systems under some conditions.


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## Body

## 1. Introduction

Recently, stochastic functional differential equations with infinite delay (IDSFDEs) have attracted broad attention of many researchers. In the literature, there are two main lines of research on the IDSFDEs. On one hand, existence and uniqueness of solution are basic properties for equations. So a great number of authors have devoted themselves to this research, and thus many excellent results on the existence and uniqueness of the solutions to IDSFDEs and neutral IDSFDEs can be found in [1–6] and references cited therein. On the other hand, the study of stability and boundedness of solutions is one of the most attracting topics in the qualitative theory of differential equations because of its various applications in many areas such as physics and control theory [7, 8]. Hence, more and more researchers study them and especially focus on the stability of solutions. An important issue in stochastic analysis is whether or not random disturbance can change the qualitative properties of system, which is particularly important in control field. In most cases, people are interested in the performance of antidisturbance of system. So it is vital to seek some antidisturbance systems or present the intensity of stochastic perturbation that stable system can tolerate without losing the property of stability [9]. In recent years, many meaningful works on this topic have come out; see, for example, [10–21].Volterra integrodifferential equations (VIDEs) are widely applied in biology, ecology, medicine, physics, among other scientific areas and thus have been encountered by many researchers in numerical and theoretic analysis; see [7, 22–25]. It is well known that concrete systems are inevitably affected by external perturbations usually modeled by stochastic noise. So a great deal of attention has been paid to the research of stochastic VIDEs [9, 26, 27]. Additionally, time delay is always ubiquitous and infinite delay systems have wide applications in many fields. Hence, there is naturally an important kind of IDSFDEs, that is, stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). In practice, many applications of IDSVIDEs are greatly dependent on the stability and boundedness of their solutions. However, to the best of the authors' knowledge, few research results mentioned above focus on the stability and boundedness of IDSVIDEs, which motivates the present study. Precisely, this paper investigates in detail the problem of stability and boundedness of solutions for the following IDSVIDE: (1)dx(t)=[A(t)x(t)+∫-∞t‍C(t,s)g(x(s))ds+f(t)]dt+[∫-∞t‍B(t)x(t)+∫-∞t‍D(t,s)g(x(s))ds+f1(t)]dW(t),t≥t0,where A(t),B(t),f(t), and f1(t):ℝ→ℝ are continuous functions. C(t,s) and D(t,s):ℝ×ℝ→ℝ are also continuous. Therein, B(t),D(t,s), and f1(t) denote the intensity of disturbance to A(t),C(t,s), and f(t), respectively.Compared with the existing results in the literature, contributions of this paper are mainly as follows.(1)
Both stochastic perturbation and infinite delay are considered in the IDSVIDEs.(2)
A new Lyapunov function is constructed to derive stability and boundedness criteria for IDSVIDEs efficiently.(3)
The problem of how much the stochastic noise VIDEs with infinite delay can tolerate without losing the properties of stability and boundedness has been solved.


## 2. Preliminaries

Let(Ω,ℱ,𝔽,ℙ) be a complete probability space with a filtration 𝔽={ℱt}t≥0 satisfying the usual conditions. As usual, W(·) denotes a scalar Brownian motion defined on the space and 𝔼(·) is the mathematical expectation with respect to ℙ. Write BC((-∞,0];ℝ) as the family of bounded continuous real-valued functions ϕ defined on (-∞,0] with the norm ∥ϕ∥=sup-∞<θ≤0|ϕ(θ)|.In this paper, we suppose there exists a constantJ>0 such that |g(x)|≤J|x|2. Then by Theorem 5.2.7 of [28], there exists a unique global solution x(t) to (1) if A(t) and B(t) are bounded. For more details, readers can see [29, 30]. Here and in the rest of the paper, write the solutions with the initial condition xt0=ϕ∈BC((-∞,0];ℝ) as x(t;t0,ϕ). Throughout this paper, unless otherwise specified, we use the following Lyapunov function:
(2)V(x,t)=x22+k∫-∞t‍∫t+∞‍|C(u,s)||g(x(s))|duds.
For any M>0, define two stopping times:
(3)τM=inf{t≥t0:|x(t)|≤M},τM=inf{t≥t0:|x(t)|≥M}.In order to study the stability and boundedness of solutions to (1), we state the following assumptions:(K1)
There exist nonnegative continuous functionsh:ℝ-→ℝ+ and F:ℝ→ℝ, such that
(4)∫-∞0‍h(s)ds=l<∞,|C(t,s)|≤F(t)h(s-t),∫t+∞‍F(u)du≤a<∞,limt→∞∫t+∞‍F(u)du=0.(K2)
For anyα∈(0,1), there exists k>1 such that
(5)A(t)+B2(t)+kJ∫t+∞‍|C(u,t)|du≤0,m(t)D(t,s)<k-αsup|x|≤α|g(x)||C(t,s)|,where m(t):=∫-∞t‍|D(t,s)|ds.(K3)
For anyα∈(0,1), there exist k>1 and δ>0 such that
(6)A(t)+B2(t)+kJ∫t+∞‍|C(u,t)|du<-δ,m(t)D(t,s)<k-αsup|x|≤α|g(x)||C(t,s)|,where m(t) is defined in (K2).Remark 1.
Conditions (K2) and (K3) are the conditions about the intensity of perturbations. Note that constantk in equality (2) is the same as the one which appeared in conditions (K2) and (K3).


## 3. Stability of Solutions of IDSVIDEs

In order to consider the stability of (1), without loss of generality, we suppose that f(t)=f1(t)=0 and g(0)=0 hold in this section. Hence, (1) has the trivial solution x(t)≡0. First, we introduce three kinds of definitions about stability of solutions to (1). It is easy to see that these definitions are a strict generalization of deterministic cases.Definition 2 (see [28]).
(
1
) The trivial solution of (1) is said to be stochastically stable if for every pair ε∈(0,1) and r>0, there exists a δ=δ(ε,r,t0)>0 such that
(7)ℙ(|x(t;t0,ϕ)|<r,t≥t0)≥1-ε,
whenever ∥ϕ∥<δ. Additionally, it is said to be stochastically uniformly stable if δ is independent of t0.
(
2
) The trivial solution of (1) is said to be stochastically asymptotically stable if it is stochastically stable and, moreover, for any ε∈(0,1), there is δ0=δ0(ε,t0)>0 such that
(8)ℙ(limt→∞x(t;t0,ϕ)=0)≥1-ε,
whenever ∥ϕ∥<δ0.
(
3
) The trivial solution of (1) is said to be stochastically globally asymptotically stable if it is stochastically stable and, moreover, for all ϕ∈BC((-∞,0];ℝ), it follows that
(9)ℙ(limt→∞x(t;t0,ϕ)=0)=1.In the following, we will apply the Lyapunov-Krasovskii functional approach to delve into some sufficient criteria, under which the trivial solution to (1) is stochastically stable, stochastically asymptotically stable, and stochastically globally asymptotically stable, respectively. The Itô formula used in this paper can be seen in [28].Theorem 3.
Suppose that (K1) and (K2) hold. Then the trivial solution of (1) is stochastically uniformly stable.Proof.
For anyε∈(0,1) and r∈(0,1), choose δ sufficiently small for δ<r2ε/(1+2kJla). For any given ϕ∈BC((-∞,0];ℝ) satisfying ∥ϕ∥≤δ, we can get from (2) that
(10)ℒV(x(s),s)=Vt(x(s),s)+Vx(x(s),s)×[A(s)x(s)+∫-∞s‍C(s,u)g(x(u))du]+12Vxx(x(s),s)[B(s)x(s)+∫-∞s‍D(s,u)g(x(u))du]2=k∫s+∞‍|C(u,s)||g(x(s))|du-k∫-∞s‍|C(s,u)||g(x(u))|du+A(s)x2(s)+x(s)∫-∞s‍C(s,u)g(x(u))du+12[B(s)x(s)+∫-∞s‍D(s,u)g(x(u))du]2≤(kJ∫s+∞‍|C(u,s)|du+A(s)+B2(s))x2(s)-(k-|x(s)|)∫-∞s‍|C(s,u)||g(x(u))|du+[∫-∞s‍D(s,u)g(x(u))du]2.
Then it follows that, for any t≥t0,
(11)𝔼V(x(t∧τr),t∧τr)=V(ϕ(t0),t0)+𝔼∫t0t∧τr‍[[∫-∞s‍D(s,u)g(x(u))du]2(kJ∫s+∞‍|C(u,s)|du+A(s)+B2(s))x2(s)-(k-|x(s)|)∫-∞s‍|C(s,u)||g(x(u))|du+[∫-∞s‍D(s,u)g(x(u))du]2]ds≤V(ϕ(t0),t0)+𝔼∫t0t∧τr‍[(kJ∫s+∞‍|C(u,s)|du+A(s)+B2(s))x2(s)+∫-∞s‍[(sup|x|≤r|g(x)|)-(k-r)|C(s,u)|+(sup|x|≤r|g(x)|)m(s)|D(s,u)|]×|g(x(u))|du(kJ∫s+∞‍|C(u,s)|du+A(s)+B2(s))]ds≤V(ϕ(t0),t0).
Then by a straightforward computation we obtain that
(12)V(ϕ(t0),t0)=ϕ2(t0)2+k∫-∞t0‍∫t0+∞‍|C(u,s)||g(x(s))|duds≤(12+kJla)∥ϕ∥2<r22ε.
From the definition of τr, it follows that
(13)𝔼V(x(t∧τr),t∧τr)≥𝔼(I{τr<t}V(x(τr),τr))≥r22ℙ(τr<t).
So
(14)ℙ(τr<t)<ε.
Let t→∞. We deduce that
(15)ℙ(τr<∞)≤ε,
that is,
(16)ℙ(|x(t;t0,ϕ)|≤r,t≥t0)>1-ε.
For any r1∈[1,+∞), we can have
(17){ω∣|x(t;t0,ϕ)|≤r,t≥t0}⊆{ω∣|x(t;t0,ϕ)|≤r1,t≥t0},
which implies that
(18)ℙ(|x(t;t0,ϕ)|≤r1,t≥t0)≥ℙ(|x(t;t0,ϕ)|≤r,t≥t0)>1-ε.
This completes the proof.Remark 4.
IfB(t)=D(t,s)=0 in (1), then we can get the corresponding disturbance free system. Theorem 3 really tells us that stochastic perturbation cannot disturb the stability of original deterministic system if the noisy intensity satisfies (K2).Lemma 5.
Assume that (K1) and (K2) hold. Then for anyε∈(0,1),α>0, there is R(ε,α)>0 such that
(19)ℙ(|x(t;θ*,ϕ)|≤R,t≥θ*)>1-ε,
for any θ*≥t0,∥ϕ∥<α, a.s.Proof.
For any givenε∈(0,1),α>0, and ϕ∈BC((-∞,0];ℝ) satisfying ∥ϕ∥≤α, choose R(ε,α) sufficiently large such that
(20)R2>α2(1+2kJla)ε.
Denote x(t)≜x(t;θ*,ϕ). By using a similar argument as in Theorem 3, we have that, for any t≥θ*,
(21)𝔼V(x(t∧τR),t∧τR)≤𝔼V(ϕ(θ*),θ*)=𝔼ϕ2(θ*)2+k𝔼∫-∞θ*‍∫θ*+∞‍|C(u,s)||g(x(s))|duds≤𝔼ϕ2(θ*)2+kJla𝔼∥ϕ∥2≤(12+kJla)𝔼∥ϕ∥2.
Making use of the definition of V(x,t) yields
(22)𝔼V(x(t∧τR),t∧τR)≥𝔼(I{τR<t}V(x(τR),τR))≥R22ℙ(τR<t).
Combining this and (21), we have
(23)ℙ(τR<t)≤(1+2kJla)𝔼∥ϕ∥2R2<ε.
Let t→∞. Then
(24)ℙ(τR<∞)≤ε,
which can imply our desired result
(25)ℙ(|x(t;θ*,ϕ)|≤R,t≥θ*)>1-ε.Theorem 6.
Suppose that (K1) and (K3) hold. Then the trivial solution is stochastically asymptotically stable and stochastically globally asymptotically stable.Proof.
(I) The proof of stochastic asymptotic stability.
For anyθ*>t0, ϕ*={ϕ*(θ):-∞<θ≤0} is ℱθ*-adapted BC((-∞,0];ℝ)-valued random variable such that 𝔼∥ϕ*(θ)∥<∞. Let x(t;θ*,ϕ*) denote the solution with the initial value (θ*,ϕ*). From Theorem 3, the trivial solution is stochastically uniformly stable. Hence, for any given δ1>0,ε∈(0,1), there exists δ(ε,δ1)>0, such that
(26)ℙ(ω:|x(t;t0,ϕ)|<δ1,t≥t0)≜ℙ(A)>1-ε,
whenever ∥ϕ∥<δ(ε,δ1). Fix ϕ∈BC((-∞,0];ℝ) such that ∥ϕ∥<δ(ε,δ1). For any given ω*∈A,σ≥t0, define
(27)ϕσ(t,ω)=x(t;t0,ϕ)I(t,ω)∈(-∞,σ]×A+x(t;t0,ϕ,ω*)I(t,ω)∈(-∞,σ]×(Ω-A).
So Lemma 5 shows that, for δ1 above and any ε1∈(0,1), there is H(ε1,δ1) such that
(28)ℙ(|x(t;σ,ϕσ)|≤H,t≥σ)>1-ε14.
If there exists K>t0 such that
(29)ℙ(ω∈A:|x(t;t0,ϕ)|=0,t≥K)=ℙ(A)>1-ε,
then the trivial solution of system (1) is stochastically asymptotically stable. If not, there are a sequence γk and increasing sequence τk such that limk→∞τk=+∞ and
(30)ℙ(ω∈A:|x(τk;t0,ϕ)|≠0)=γk>0,k=1,2,…
Define
(31)Skβ=(ω∈A:|x(τk;t0,ϕ)|>β),k=1,2,…
It is not difficult to see that
(32)limβ→0ℙ(Skβ)=γk,k=1,2,…
Hence, there are β0>0 and positive sequence γk0 such that(33)ℙ(ω∈A:|x(τk;t0,ϕ)|>β0)≜ℙ(Aτk)=γk0>0,k=1,2,…
If for any β1∈(0,β0), the following holds:
(34)ℙ(ω∈A:limsupt→∞|x(t;t0,ϕ)|<β1)>1-ε,
which can show the trivial solution of system (1) is stochastically asymptotically stable. If not, there is β2∈(0,β0) such that
(35)ℙ(ω∈A:limsupt→∞|x(t;t0,ϕ)|<β2)≜ℙ(Bβ2)≤1-ε.
Obviously, ℙ(A-Bβ2)>0, Aτk⊆A-Bβ2, and for any ω∈A-Bβ2,
(36)limsupt→∞|x(t;t0,ϕ)|≥β2>0.
For any given B<β2,α<B such that 2α2/B2<ε1/4. Choose N sufficiently large such that, for any t>τN,
(37)∫t+∞‍F(u)du<α22klJH2.
For any given ω*∈AτN, define
(38)ϕτN(t,ω)=x(t;t0,ϕ)I(t,ω)∈(-∞,τN]×AτN+x(t;t0,ϕ,ω*)I(t,ω)∈(-∞,τN]×(Ω-AτN).
So (28) can tell us that
(39)ℙ(ω:|x(t;τN,ϕτN)|<H,t≥τN)>1-ε14,(40)x(t;τN,ϕτN)=x(t;t0,ϕ),(t,ω)∈(-∞,+∞)×AτN.
From here, we let x(t)≜x(t;τN,ϕτN),t≥τN, and let V(x,t) be the same as (2).
By The Itô formula and (K3), for anyt≥τN,
(41)𝔼V(x(t∧τα∧τH),t∧τα∧τH)=V(ϕτN(τN),τN)+𝔼∫τNt∧τα∧τH‍ℒV(x(s),s)ds≤V(ϕτN(τN),τN)-δα2𝔼(t∧τα∧τH-τN),
where ℒV(x(s),s) is the same as (10). So,
(42)(t-τN)ℙ{τα∧τH≥t}≤𝔼(t∧τα∧τH-τN)≤V(ϕτN(τN),τN)δα2.
Letting t→∞ can yield
(43)ℙ{τα∧τH<∞}=1.
Clearly, it follows from (39) that ℙ(τH<∞)≤ε1/4. Therefore,
(44)1=ℙ(τα∧τH<∞)≤ℙ(τα<∞)+ℙ(τH<∞)≤ℙ(τα<∞)+ε14.
Hence,
(45)ℙ(τα<∞)≥1-ε14.
Choose θα sufficiently large such that
(46)ℙ(τα<θα)≥1-ε12.
Then
(47)ℙ(τα<τH∧θα)≥ℙ({τα<θα}∩{τH=∞})≥ℙ(τα<θα)-P(τH<∞)≥1-3ε14.
Define two stopping times:
(48)σ={τα,τα<τH∧θα;∞,otherwise,τB=inf{t>σ:|x(t)|≥B}.
So for any t≥θα,
(49)𝔼V(x(τB∧t),τB∧t)≤𝔼V(x(σ∧t),σ∧t).
Note that if ω∈{τα≥τH∧θα}, then
(50)V(x(τB∧t),τB∧t)=V(x(σ∧t),σ∧t).
Consequently,
(51)𝔼(I{τα<τH∧θα}V(x(τB∧t),τB∧t))≤𝔼(I{τα<τH∧θα}V(x(τα),τα)).
Noting {τB<t}⊆{τα<τH∧θα}, it yields that
(52)𝔼(I{τα<τB∧θα}V(x(τB∧t),τB∧t))≥(𝔼(I{τα<τB∧θα}V(x(τB∧t),τB∧t))∣τB<t)ℙ(τB<t)≥B22ℙ(τB<t).
In view of the definition of V(x,t) and condition (K1),
(53)𝔼(I{τα<τH∧θα}V(x(τα),τα))≤𝔼V(x(τα),τα)=𝔼[x2(τα)2+k∫-∞τα‍∫τα+∞‍|C(u,s)||g(x(s))|duds]≤𝔼[α22+kJH2∫τα+∞‍∫-∞τα‍F(u)h(s-u)dsdu]≤α22+kJlH2∫τα+∞‍F(u)du<α2,
which, in conjunction with (51) and (52), yields that
(54)ℙ(τB<t)<2α2B2<ε14.
Letting t→∞, it yields that
(55)ℙ(τB<∞)≤ε14.
Hence,
(56)ℙ({σ<∞}∩{τB=∞})≥ℙ(τα<τH∧θ)-ℙ(τB<∞)≥1-ε1.
This implies immediately that
(57)ℙ(limsupt→∞|x(t)|≤B)≥1-ε1.
At last, from the arbitrariness of B, it must be
(58)ℙ(limsupt→∞|x(t)|=0)≥1-ε1.
Since ε1 is arbitrary, we then obtain that
(59)ℙ(limsupt→∞|x(t)|=0)=1.
Hence, from (40), there is ω∈AτN⊆A-Bβ2 such that
(60)limsupt→∞|x(t;t0,ϕ)|=0.
But this is in contradiction with (36), and thus (34) must hold and this completes the proof.
(II) The proof of stochastic global asymptotic stability.
Give anyε∈(0,1), ϕ∈BC((-∞,0];ℝ). Let H be sufficiently large such that
(61)inf|x|>H,t≥t0V(x,t)≥4V(ϕ(t0),t0)ε.
Then we can easily have
(62)𝔼V(x(t∧τH),t∧τH)<𝔼V(ϕ(t0),t0).
From the assumption of H,
(63)𝔼V(x(t∧τH),t∧τH)≥𝔼(I{τH<t}V(x(τH),τH))≥4V(ϕ(t0),t0)εℙ(τH<t).
So
(64)ℙ(τH<t)<ε4.
Let t→∞. we could get
(65)ℙ(τH<∞)<ε4,
namely,
(66)ℙ(|x(t;t0,ϕ)|≤H)>1-ε4.
Arguing as part (I), we obtain that
(67)ℙ(limt→∞x(t;t0,ϕ)=0)>1-ε.
From the arbitrariness of ε, the trivial solution of (1) is stochastically globally asymptotically stable, which ends the proof.In the last part of this section, we give two examples for better understanding the stability theorems above.Example 7.
Consider the following IDSVIDE:(68)dx(t)=[(-2e-t-1)x(t)+∫-∞t‍es-2tx2(s)ds]dt+[∫-∞t‍e-t/2x(t)+∫-∞t‍es-γtx2(s)ds]dW(t),t∈[0,+∞),
where γ>2 is a constant.Obviously,|g(x)|=x2, so J=1. Let h(s)=es, F(t)=e-t. Then we have ∫-∞0‍h(s)ds=1<∞ and
(69)∫t+∞‍F(u)du=e-t≤1<∞,limt→∞∫t+∞‍F(u)du=0;|C(t,s)|=es-2t=es-te-t=F(t)h(s-t).
For any α∈(0,1), let k=2. Easily to get
(70)B2(t)=e-t,A(t)+kJ∫t+∞‍|C(u,t)|du=-e-t-1,m(t)D(t,s)=e-2γt+t+s,k-αsup|x|≤α|g(x)||C(t,s)|>es-2t.
Therefore,
(71)B2(t)<-(A(t)+kJ∫t+∞‍|C(u,t)|du),m(t)D(t,s)<k-αsup|x|≤α|g(x)||C(t,s)|.
Hence all the conditions of Theorem 3 have been verified, and it follows that the trivial solution of (1) is stochastically uniformly stable.Example 8.
Consider an IDSVIDE as follows:(72)dx(t)=[∫-∞t‍(-log2(x2(t)+1)-2)x(t)+∫-∞t‍e-t(t-s+2)2sin(x2(s))ds]dt+[∫-∞t‍log2(x2(t)+1)x(t)+∫-∞t‍eβs-γtsin(x2(s))ds]dW(t),t∈[0,+∞),
where β>4,γ>β+1 are constants.Obviously,|g(x)|=|sinx2|≤x2, so we can let J=1. Let h(s)=1/(1-s)2, F(t)=e-t. Then we have ∫-∞0‍h(s)ds=1<∞ and
(73)∫t+∞‍F(u)du=e-t≤1<∞,limt→∞∫t+∞‍F(u)du=0;|C(t,s)|=e-t(t-s+2)2≤e-t(t-s+1)2=F(t)h(s-t).
Additionally, for any α∈(0,1), let k=2 and δ=1/2. Then it is not difficult to check that
(74)B2(t)=log2(x2(t)+1)<log2(x2(t)+1)+12=-(A(t)+2∫t+∞‍1(u-t+2)2du)-12≤(A(t)+kJ∫t+∞‍|C(u,t)|du)-δ.
Consider function f(u)=eβu(u-2)2-β,u<0. Clearly, f(u) is differentiable and
(75)f′(u)=eβu(u-2)(β(u-2)+2)>0,f(0)=4-β<0.
So f(u)<0, that is, eβu/β<1/(u-2)2. Hence,
(76)sup|x|≤α(sinx2)m(t)D(t,s)≤D(t,s)∫-∞t‍|D(t,s)|ds=1βe(2β-2γ)teβs-βt<e(2β-2γ)t(s-t-2)2<|C(t,s)|.
Hence, all of the conditions of Theorem 6 are satisfied and we can assert that the trivial solution of (1) is stochastically asymptotically stable and stochastically globally asymptotically stable.


## 4. Boundedness of Solutions of IDSVIDEs

In the section, we begin with three types of definitions about boundedness for solutions of (1).Definition 9.
(
1
) A solution x(t) of (1) is stochastically bounded, if, for any ε∈(0,1), ϕ∈BC((-∞,0];ℝ), there exists R(ε,ϕ)>0, such that
(77)ℙ(supt≥t0|x(t;t0,ϕ)|≤R)>1-ε.
(
2
) The solutions of (1) are stochastically equibounded, if, for any ε∈(0,1), α>0, and ϕ∈BC((-∞,0];ℝ), there exists R(ε,α)>0, such that
(78)ℙ(supt≥t0|x(t;t0,ϕ)|≤R,∥ϕ∥<α)>1-ε.
(
3
) Fix β>0; the solutions of (1) are stochastically ultimately bounded for bound β, if, for any ε∈(0,1), ϕ∈BC((-∞,0];ℝ), there exists a T(ε,ϕ)≥t0, such that
(79)ℙ(supt≥T(ε,ϕ)|x(t;t0,ϕ)|≤β)>1-ε.Now we apply the Lyapunov-Krasovskii method to give the sufficient conditions, under which the solutions of (1) are stochastically bounded, stochastically equibounded, and stochastically ultimately bounded, respectively.Theorem 10.
Suppose that (K1) and (K2) hold. Iff(t)=0, then the solutions to (1) are stochastically bounded and stochastically equibounded.Proof.
(I) Proof of stochastic boundedness.
Letε∈(0,1), ϕ∈BC((-∞,0];ℝ) be arbitrary. Choose H sufficiently large such that inf|x|>H,t≥t0V(x,t)≥V(ϕ(t0),t0)/ε. By the Itô formula, for any t≥t0, we can obtain that
(80)V(x(t∧τH),t∧τH)=V(ϕ(t0),t0)+∫t0t∧τH‍ℒV(x(s),s)ds+∫t0t∧τH‍Vx(x(s),s)×[∫-∞s‍B(s)x(s)+∫-∞s‍D(s,r)g(x(r))dr+f1(s)]dW(s),a.s.
Taking expectation on both sides of (80), it follows by using (10) that
(81)𝔼V(x(t∧τH),t∧τH)≤V(ϕ(t0),t0).
From the assumption H, we derive that
(82)𝔼V(x(t∧τH),t∧τH)≥𝔼(I{τH<t}V(x(τH),τH))≥V(ϕ(t0),t0)εℙ(τH<t).
We therefore must have
(83)ℙ(τH<t)≤ε.
Letting t→∞, we can obtain that
(84)ℙ(τH<∞)≤ε.
That is,
(85)ℙ(supt≥t0|x(t)|≤H)>1-ε.
The proof of stochastic boundedness is complete.
(II) Proof of stochastic equiboundedness.
Letε∈(0,1) and α>0 be arbitrary. Give any ϕ∈BC((-∞,0];ℝ), such that ∥ϕ∥≤α. Choose R such that R>α2(1+2kJla)/2ε. By applying the Itô formula, for any t≥t0,
(86)V(x(t∧τR),t∧τR)=V(ϕ(t0),t0)+∫t0t∧τR‍ℒV(x(s),s)ds+∫t0t∧τR‍Vx(x(s),s)×[B(s)x(s)+∫-∞s‍D(s,r)g(x(r))dr+f1(s)∫-∞s‍]dW(s),a.s.
Taking the expectation for both sides of (86), we have
(87)𝔼V(x(t∧τR),t∧τR)≤V(ϕ(t0),t0).
In view of
(88)V(ϕ(t0),t0)=ϕ2(t0)2+k∫-∞t0‍∫t0+∞‍|C(u,s)||g(x(s))|duds≤(12+kJla)∥ϕ∥2,𝔼V(x(t∧τR),t∧τR)≥𝔼(I{τR<t}V(x(τR),τR))≥Rℙ(τR<t).
Consequently,
(89)ℙ(τR<t)≤(1+2kJla)∥ϕ∥22R≤α2(1+2kJla)2R<ε.
Letting t→∞, it yields that
(90)ℙ(τR<∞)≤ε,
that is,
(91)ℙ(supt≥t0|x(t)|≤R,∥ϕ∥<α)>1-ε.
This completes the proof.Remark 11.
Notice that, when conditions (K1) and (K2) hold, the solutions of stochastic system (1) are stable and bounded. That is, environmental noise (in the sense of Itô) cannot disturb stability and boundedness of solutions for some systems if noisy intensity satisfies (K2). Hence, we can construct some anti-interference systems in practice.Theorem 12.
Suppose that (K1) and (K3) hold. Iff(t)=0, then the solutions to (1) are stochastically ultimately bounded.Proof.
Giveβ>0. For any ε∈(0,1), ϕ∈BC((-∞,0];ℝ). From Theorem 10, there exists H>0, such that
(92)ℙ(|x(t)|≤H,t≥t0)≥1-ε4.
Choose α<|ϕ(t0)|∧β, and define four stopping times as follows:
(93)τα=inf{t≥t0:|x(t)|≤α},τH=inf{t≥t0:|x(t)|≥H},τβ=inf{t>σ:|x(t)|≥β},σ={τα,τα<τH∧θα;∞,otherwise.
From here, we can show in the same way as in the proof of Theorem 6 that we could choose sufficiently small α, such that 2α2/H2<ε and
(94)H22ℙ(τβ<t)≤𝔼(I{τα<τH∧θα}V(x(τβ∧t),τβ∧t))≤𝔼(I{τα<τH∧θα}V(x(τα),τα))≤α2.
Then it follows that
(95)ℙ(τβ<t)<2α2H2<ε.
After letting t→∞, we have
(96)ℙ(τβ<∞)≤ε.
Let T(ε,ϕ)=τα. Then the above inequality is equivalent to
(97)ℙ(supt≥T|x(t)|≤β)>1-ε.
The proof is complete.Remark 13.
Obviously, we can verify that the solutions of IDSVIDE (68) are stochastically bounded and stochastically equibounded. And the solutions of IDSVIDE (72) are stochastically ultimately bounded.


## 5. Conclusions

Throughout this paper, by combining the Lyapunov-Krasovskii method, we have obtained various kinds of sufficient stability and boundedness criteria, where the ranges of noisy intensity that stable and bounded systems can tolerate without losing the properties of stability and boundedness are presented, respectively. These sufficient conditions are very necessary for us to verify stability and boundedness of stochastic Volterra integrodifferential equations with infinite delays. Moreover, the conditions obtained in this paper can also help us to construct some antidisturbance systems in the applications. In addition, two examples have been given to illustrate our theoretical results.



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