# Stability of a Class of Hybrid Neutral Stochastic Differential Equations with Unbounded Delay

Authors: Boliang Lu; Ruili Song
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

This paper studies the stability of hybrid neutral stochastic differential equations with unbounded delay. Some novel exponential stability criteria and boundedness conditions are established based on the generalized Itô formula and Lyapunov functions. The factore-εδ(t) is used to overcome the difficulties caused by the unbounded delay δ(t) effectively. In particular, our results generalize and improve some previous stability results from bounded delay to unbounded delay conditions. Finally, an example is presented to demonstrate the effectiveness of the proposed results.


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## Body

## 1. Introduction

With the deep development of the research on complex systems in recent years, much attention has been paid to the influence of random factors on the power system. Hence, the stochastic differential equations (SDEs) come to play important roles in many fields, such as molecular physics, population genetics, and some other branches of science (see [1–3]).Practically, there are many systems depending on both present and past states, as well as the changing rate. To model this kind of systems primely, the delay systems and neutral stochastic differential equations (NSDEs) with time-dependent delay are studied widely. Recently, there have appeared many interesting results in the literature. For example, [4–6] studied the global asymptotic stability results and applications of time-varying delay systems in neural networks (NNs), memristive neural networks (MNNs), and chaotic Lure systems (CLSs). Zhu and Cao [7] investigated the global asymptotic stability for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays. Liu and Zhu [8] studied mean square stability of two classes of theta method for NSDEs with bounded delay. Ma and Xi [9] established a moderate deviation principle for NSDEs driven by Poisson random measure and time delay.On the other hand, some random abrupt changes generally exist in practical systems because of the change of internal and external environment. SDEs with Markovian switching (SDEwMs) are powerful tools for describing systems that encounter abrupt changes in structure. It is inspiring that many important results have been reported in the literature. For example, Zhu et al. [10] investigated the problem of robust exponential stability for a class of stochastically nonlinear jump systems with mixed time delays. Zhu [11] investigated the pth moment exponential stability for a class of impulsive stochastic functional differential equations with Markovian switching (SFDEwMs). Furthermore, Zhu [12] studied the Razumikhin-type theorem for a class of SFDEwMs with Lévy noise, which was a breakthrough of the traditional Brownian motion. Reference [13] discussed the asymptotic stability of SDEwMs. Also, in recent years many researchers pay more effort to applying Markovian switching into the neural networks that has various applications (e.g., see [14–16]). Besides, a few stability studies on the NSDEs with Markovian switching (NSDEwMs) can be found in [17–20].Above references just focus on bounded time delay, leaving many practical systems unanswered where the delay terms are usually required to be unbounded. For example, in the neural network model, stochastic unbounded variable delay differential system must be considered to model transmission and transformation of the signal in a better fashion (see [19]). Some related works on unbounded delay can be found in [20–24]. References [22, 24] investigated the existence and uniqueness, as well as the pathwise stability of the global solutions to SDEs and NSDEs with unbounded delay, respectively. Reference [23] studied the stability of SFDEs with unbounded delay. In Section 4.2 of [21], the stability and boundedness of nonlinear hybrid SDEs were discussed when the delay function δ(t) was given as θt(0<θ<1). Particularly, [20] presented the existence and uniqueness of the exact solution for a class of NSDEwMs with unbounded delay and also showed the pth moment exponential stability and almost sure exponential stability results under the bounded delay condition and established the Euler-Maruyama method under both cases. As far as we know, few works have been devoted to study the stability of neutral stochastic hybrid systems with unbounded delay.Motivated by the above discussion, this paper aims to develop thepth moment exponential stability and almost sure exponential stability criteria of the solution of NSDEwMs with unbounded delay after showing the corresponding existence and uniqueness theorem. Our methods rely upon the generalized Itô formula and Lyapunov functions. Besides, the factor e-εδ(t) is used to overcome the key challenges emerging from the neutral term and unbounded delay features. The contributions of this paper are twofold. The first one is that we improve the delay function δt=θt(0<θ<1) in [21] to the general unbounded delay and add the neutral term. The other one is that we greatly generalize and advance the pth moment exponential stability and almost sure exponential stability results in [20] where the delay term was bounded. It should be pointed out that we use the Euler-Maruyama method which was investigated in [20] to simulate a numerical example in this study.The remainder of this paper is organized as follows. In Section2, we define some notations and give four assumptions with respect to the system of the hybrid neutral stochastic differential equations with unbounded delay. In Section 3, we obtain our results through three theorems. A numerical example is proposed to demonstrate the results in Section 4. Finally, in Section 5, we conclude this paper with some general remarks.


## 2. Models, Notations, and Assumptions

We use the following notations throughout this paper without specification.Let(Ω,F,Ftt≥0,P) be a complete probability space with a filtration Ftt≥0 satisfying the usual conditions, which means that it is increasing and right continuous, and F0 contains all P-null sets. Let {w(t)=(w1(t),w2(t),…,wm(t))T,t≥0} be an m-dimensional standard Brownian motion defined on the probability space (Ω,F,{Ft}t≥0,P). For any x∈Rd, let x stand for the Euclidean norm. If A is a matrix, demote its trace norm by A2=trace(ATA), where AT is the transpose. For a given τ>0, denote by C([-τ,0];Rd) the family of continuous function φ:[-τ,0]→Rd with the supremum norm φ=sup-τ≤θ≤0φθ. Also, denote by CF0b([-τ,0];Rd) the family of bound, F0-measurable, C([-τ,0];Rd)-valued random variables and R+=[0,∞).Let{r(t),t≥0} be a right-continuous Markovian chain on the probability space (Ω,F,{Ft}t≥0,P) taking values in a finite states space S={1,2,…,N} with generator Γ=(γij)N×N given by (1)Prt+Δ=j∣rt=i=γijΔ+oΔ,ifi≠j,1+γijΔ+oΔ,ifi=j,where Δ>0 and γij≥0 is the transition rate from i to j if i≠j, while γii=-∑i≠jγij. Assume that the Markov chain {r(t)}t≥0 is independent of the Brownian motion {w(t)}t≥0.The model considered in this paper is the following hybrid neutral stochastic differential equation with unbounded delay:(2)dxt-uxt-δt,t,rt=fxt,xt-δt,t,rtdt+gxt,xt-δt,t,rtdwt,t≥0with initial conditions (3)x0=φ=φθ:θ∈-δ0,0∈CF0b,r0=i0∈S,where the functions (4)f:Rd×Rd×R+×S⟶Rd,g:Rd×Rd×R+×S⟶Rd×m,u:Rd×R+×S⟶Rdare all Borel-measurable and x(t) is a d-dimensional state process. The unbounded delay function δ:R+→R+ is Borel-measurable. It is necessary to say that model (2) was investigated in [20] to study the existence and uniqueness of the solution under the unbounded delay, as well as the pth moment exponential stability and almost sure exponential stability results under the bounded delay condition. Next we will further establish the pth moment exponential stability and almost sure exponential stability criteria of model (2) when the delay function is unbounded. Besides, the following Assumption 4 we used is quite different from the assumption A2 in [20].Denote byC(Rd×R+;R+) the family of all continuous functions from Rd×R+ to R+. Let C2,1(Rd×R+×S;R+) denote the family of all continuous nonnegative functions V(x,t,i) from Rd×R+×S to R+ such that, for each i∈S, they are continuously twice differentiable in x and once in t.For a givenV∈C2,1(Rd×R×S;R+), we define an operator LV:Rd×Rd×R+×S→R by (5)LVx,y,t,i=Vtx-uy,t,i,t,i+Vxx-uy,t,i,t,ifx,y,t,i+12tracegTx,y,t,iVxxx-uy,t,i,t,igx,y,t,i+∑j=1NγijVx-uy,t,i,t,j,where (6)Vtx,t,i=∂Vx,t,i∂t,Vxx,t,i=∂Vx,t,i∂x1,∂Vx,t,i∂x2,…,∂Vx,t,i∂xd,Vxxx,t,i=∂2Vx,t,i∂xi∂xjd×d.By the generalized Itô formula, we have(7)Vxt-uxt-δt,t,rt,t,rt-Vx0-ux0,0,r0,0,r0=∫0tLVxs,xs-δs,s,rsds+∫0tVxxs-uxt-δs,s,rs,s,rsgxs,xs-δs,s,rsdws+∫0t∫RVxs-uxs-δs,s,rs,s,i0+hrs,l-Vxs-uxs-δs,s,rs,s,rsμds,dl,where the function h:S×R→R is defined by (8)hi,y=j-i,ify∈Δij,0,otherwise,and μ(ds,dl)=ν(ds,dl)-μ(dl)ds is a martingale measure while ν(dt,dy) is a Poisson random measure with intensity dt×μ(dy), in which μ is a Lebesgue measure on R. The definition of Δij and more related details can be found in [25].Now we give the following primary inequalities.Proposition 1.
For anya,b,p∈R+ and η∈(0,1), the following inequalities hold. (9)a+bp≤ap+bp,0<p≤1,(10)a+bp≤1+ηp-1ap+η1-pbp,p>1.The detailed descriptions and proof of Proposition1 can be found in [25].Nevertheless, additional assumptions will be considered to achieve the further conclusions.Assumption 2 (the local Lipschitz condition).
For each positive integerh, there exists a positive constant Kh such that for all x1,x2,y1,y2∈Rd with x1∨x2∨y1∨y2≤h and for all t∈R+, i∈S, (11)fx1,y1,t,i-fx2,y2,t,i2∨gx1,y1,t,i-gx2,y2,t,i2≤Khx1-x22+y1-y22.Assumption 3.
The delay functionδ:R+→R+ is differentiable and there exists a constant δ- satisfying (12)δ′t≤δ-<1.Letδ∗(t)=t-δ(t), t∈R+; then δ∗′(t)≥1-δ->0, so δ∗(t) is an increasing function of t and t-δ(t)≥-δ(0), t∈R+.Assumption 4.
There are a functionV∈C2,1(Rd×R+×S;R+), nonnegative constants λ1, λ2, λ3, and positive constants p, c1, c2 with λ2>λ3, such that (13)c1xp≤Vx,t,i≤c2xp,and there also exists ε>0 such that (14)LVx,y,t,i≤λ1-λ2Vx,t,i+λ31-δ-e-εδtVy,t-δt,i,for x,y∈Rd, t∈R+, and i∈S.Assumption 5.
For the sameε in Assumption 4, there is a constant β∈(0,1) such that (15)uxt-δt,t,rt≤βe-εδt/pxt-δt,for x,y∈Rd, t≥0, and i∈S.Moreover, inequality (15) implies that (16)uxt,t,i≤βxt.


## 3. Main Results

In this section, we will obtain thepth moment and almost sure exponential stability criteria of the solution of model (2) under Assumptions 2–5. In the following theorem, we will obtain the existence and uniqueness for the global solution of model (2).Theorem 6.
Let Assumptions2–5 be satisfied; then, for initial condition (3), there exists a unique global solution x=x(t); t≥0 of model (2) with the following properties hold: (17)Ext-uxt-δt,t,rtp<∞,t≥0,(18)limsupt→∞⁡1t∫0tExspds≤λ1c2λ2-λ3.Furthermore, if λ1=0, then the solution x obeys that (19)∫0∞Extpdt≤1c1λ2-λ3Ex0-ux-δ0,0,r0p+λ3E∫-δ00xspds,(20)∫0∞xtpdt<∞a.s.Proof.
It is well-known that, for a given initial condition (3), Assumption 2 supports the existence and uniqueness of maximal local solution x(t) of (2) on t∈[0,τe), where τe is the explosion time.
Sincex0 is bounded, there is a positive constant k0 such that x0=x0<k0. For each integer k≥k0, define the stopping time (21)τk=inf⁡t∈0,τe:xt≥k,inf⁡⌀=∞.Obviously, τk is increasing as k→∞. Define τ∞=limk→∞τk, where τ∞≤τe,a.s.
In order to show that the solutionx(t),t∈[0,∞), does not explode in finite time, we need to prove that τ∞=∞a.s., which implies τe=∞a.s.
By the generalized Itô formula (7) and condition (14), we obtain that, for any t≥0, (22)dVxt-uxt-δt,t,rt,t,rt=LVxt,xt-δt,t,rtdt+dM1t≤λ1-λ2Vxt,t,rt+λ31-δ-e-εδtVxt-δt,t-δt,rt-δt+dM1t,where (23)M1t=∫0tVxxs-uxs-δs,s,rs,s,rsgxs,xs-δs,s,rsdws+∫0t∫RVxs-uxs-δs,s,rs,s,i0+hrs,l-Vxs-uxs-δs,s,rs,s,rsμds,dlis a local martingale with M1(0)=0.
For anyk>k0, integrating both sides of (22) and then taking expectation, we get that, for any t≥0, (24)EVxτk∧t-uxτk∧t-δτk∧t,τk∧t,rτk∧tτk∧t,rτk∧t≤EVx0-ux-δ0,0,r0,0,r0+λ1t-λ2E∫0τk∧tVxs,s,rsds+λ31-δ-E∫0τk∧te-εδsVxs-δs,s-δs,rs-δsds.Applying (13) to the first term on the right side of (24), we have (25)EVx0-ux-δ0,0,r0,0,r0≤c2x0-ux-δ0,0,r0p.
On the other hand, according to Assumption3 and inequality e-εδ(t)<1, we get (26)λ31-δ-E∫0τk∧te-εδsVxs-δs,s-δs,rs-δsds≤λ31-δ-E∫0τk∧tVxs-δs,s-δs,rs-δsds≤λ3E∫-δ00Vxs,s,rsds+λ3E∫0τk∧tVxs,s,rsds.Substituting (25) and (26) into (24), it follows that (27)0≤c1Ext∧τk-uxt∧τk-δt∧τk,t∧τk,rt∧τkp≤EVxt∧τk-uxt∧τk-δt∧τk,t∧τk,rt∧τk,t∧τk,rt∧τk≤c2Ex0-ux-δ0,0,r0p+λ3E∫-δ00Vxs,s,rsds+λ1t-λ2-λ3E∫0τk∧tVxs,s,rsds.Bearing in mind that λ2>λ3 in Assumption 4, we obtain(28)c1Exτk∧t-uxτk∧t-δτk∧t,τk∧t,rτk∧tp≤K1+λ1t,where K1=c2Ex0-ux-δ0,0,r0p+λ3E∫-δ(0)0V(x(s),s,r(s))ds.
Defineμk=infy≥(1-β)k,t≥0c1yp. By (16), we get (29)xτk-uxτk-δτk,τk,rτkIτk≤t≥xτk-uxτk-δτk,τk,rτkIτk≤t≥xτk-βxτk-δτkIτk≤t≥1-βk.On the basis of (28) and (29), we have (30)K1+λ1t≥c1Exτk∧t-uxτk∧t-τk∧t,τk∧t,rτk∧tp≥Ec1xτk-uxτk-δτk,τk,rτkpIτk≤t≥Einfy≥1-βk,t≥0c1ypIτk≤t=μkPτk≤t.Remarking that limx→∞c1xp→∞, then limk→∞μk=∞. Letting k→∞ in (30), we have Pτ∞≤t=0; that is, Pτ∞>t=1. For the arbitrariness of t, we know that Pτ∞=∞=1, which implies that τ∞=∞a.s. as required.
We have established the existence and uniqueness of the global solution, and now we further verify its properties.
Noting that inequality (27) yields (31)λ2-λ3E∫0τk∧tVxs,s,rsds≤K1+λ1t.Letting k→∞, we get (32)λ2-λ3∫0tEVxs,s,rsds≤K1+λ1t.Dividing both sides of (32) by (λ2-λ3)t and letting t→∞, combining with (13) we obtain the assertion (18).
Particularly, whenλ1=0, from (31), we have (33)c1λ2-λ3E∫0τk∧txspds≤K1.Dividing both sides of this inequality by (λ2-λ3), and letting t→∞, we get (34)∫0∞Extpdt≤K1c1λ2-λ3=1c1λ2-λ3Ex0-ux-δ0,0,r0p+λ3E∫-δ00xspds,which gives the desired estimate (19).
By Fubini’s theorem, we obtain(35)E∫0∞xtpdt≤K1c1λ2-λ3;this implies (36)∫0∞xtpdt<∞,a.s.which is (20).Remark 7.
The results of Theorem6 are similar to those of Theorem 1 in [20] in form, but Assumption 4 in this paper is different from A2 in [20].Theorem 8.
Let Assumptions2–5 be satisfied; then, for initial condition (3), the unique solution x of (2) has the property (37)limsupt→∞⁡Ext-uxt-δt,t,rtp≤λ1c1ε,where 0<ε<ε0, while ε0 is the unique root to the equation (38)λ2=λ3+c2ε1+βpc11-δ-.If, furthermore, λ1=0, then the solution x obeys that (39)limsupt→∞⁡ln⁡Ext-uxt-δt,t,rtpt≤-ε,(40)limsupt→∞⁡ln⁡xt-uxt-δt,t,rtpt≤-ε,a.s.Proof.
By the generalized Itô formula, for any0<ε<ε0, we have (41)deεtVxt-uxt-δt,t,rt,t,rt=eεtεVxt-uxt-δt,t,rt,t,rtdt+eεtLVxt,xt-δt,t,rtdt+eεtdM2t,where (42)M2t=∫0teεsVxxs-uxs-δs,s,rs,s,rsgxs,xs-δs,s,rsdws+∫RVxs-uxs-δs,s,rs,s,i0+hrs,l-Vxs-uxs-δs,s,rs,s,rsμds,dl is a local martingale with M2(0)=0.
For anyk≥k0, let τk be the stopping time defined in the proof of Theorem 6. Integrating both sides of (41), and then taking expectation, we get that, for any t≥0, (43)Eeεt∧τkVxt∧τk-uxt∧τk-δt∧τk,t∧τk,rt∧τk,t∧τk,rt∧τk=EVx0-ux-δ0,0,r0,0,r0+E∫0t∧τkeεsεVxs-uxs-δs,s,rs,s,rsds+E∫0t∧τkeεsLVxs,xs-δs,s,rsds.
By Assumption3, we get (44)E∫0t∧τkeεsLVxs,xs-δs,s,rsds≤E∫0t∧τkeεsλ1-λ2Vxs,s,rs+λ31-δ-e-εδsVxs-δs,s-δs,rs-δsds≤λ1εeεt-λ2E∫0t∧τkeεsVxs,s,rsds+λ31-δ-E∫0t∧τkeεse-εδsVxs-δs,s-δs,rs-δsds≤λ1εeεt-λ2E∫0t∧τkeεsVxs,s,rsds+λ3E∫-δ00eεsVxs,s,rsds+λ3E∫0t∧τkeεsVxs,s,rsds≤λ1εeεt+λ3E∫-δ00eεsVxs,s,rsds-λ2-λ3E∫0t∧τkeεsVxs,s,rsds≤λ1εeεt+c2λ2E∫-δ00eεsxspds-c1λ2-λ3E∫0t∧τkeεsxspds.
Next, we will estimate the second part of (43). Applying Proposition 1, as well as (13) and (15), we have the following expression when 0<p≤1: (45)E∫0t∧τkeεsεVxs-uxs-δs,s,rs,s,rsds≤c2E∫0t∧τkeεsεxs-uxs-δs,s,rspds≤c2E∫0t∧τkeεsεxsp+uxs-δs,s,rspds≤c2εE∫0t∧τkeεsxspds+c2εE∫0t∧τkeεsuxs-δs,s,rspds≤c2εE∫0t∧τkeεsxspds+c2εβpE∫0t∧τkeεs-δsxs-δspds≤11-δ-c2εE∫0t∧τkeεsxspds+11-δ-c2εβpE∫-δ00eεsxspds+11-δ-c2εβpE∫0t∧τkeεsxspds≤11-δ-c2ε1+βpE∫-δ00eεsxspds+11-δ-c2ε1+βpE∫0t∧τkeεsxspds.While for p>1, applying (10) and letting η=β in Assumption 5, we get (46)E∫0t∧τkeεsεVxs-uxs-δs,s,rs,s,rsds≤c2E∫0t∧τkeεsεxs-uxs-δs,s,rspds≤c2E∫0t∧τkeεsε1+βp-1xsp+1+βp-1β1-puxs-δs,s,rspds≤c2ε1+βp-1E∫0t∧τkeεsxspds+c2ε1+βp-1βE∫0t∧τkeεs-δsxs-δspds≤11-δ-c2ε1+βp-1E∫0t∧τkeεsxspds+11-δ-c2ε1+βp-1βE∫-δ00eεsxspds+11-δ-c2ε1+βp-1βE∫0t∧τkeεsxspds≤11-δ-c2ε1+βpE∫-δ00eεsxspds+11-δ-c2ε1+βpE∫0t∧τkeεsxspds.From (45) and (46), we get, for any p>0, (47)E∫0t∧τkeεsεVxs-uxs-δs,s,rs,s,rsds≤11-δ-c2ε1+βpE∫-δ00eεsxspds+11-δ-c2ε1+βpE∫0t∧τkeεsxspds.Substituting (44) and (47) into (43), we obtain that (48)Eeεt∧τkVxt∧τk-uxt∧τk-δt∧τk,t∧τk,rt∧τk,t∧τk,rt∧τk≤c2Ex0-ux-δ0,0,r0p+c21-δ-λ3+ε1+βpE∫-δ00eεsxspds+λ1εeεt-c1λ2-λ3-c2ε1-δ-1+βpE∫0t∧τkeεsxspds.Bearing in mind that ε0 is the unique root to (38), as well as (13), we see, for any ε∈(0,ε0), (49)c1Eeεt∧τkxt∧τk-uxt∧τk-δt∧τk,t∧τk,rt∧τkp≤K2+λεeεt,where (50)K2=c2Ex0-ux-δ0,0,r0p+c21-δ-λ3+ε1+βpE∫-δ00eεsxspds.Then letting k→∞, the inequality becomes (51)c1eεtExt-uxt-δt,t,rtp≤K2+λ1εeεt.Dividing both sides of (51) by eεt, and letting t→∞, it leads to (52)limsupt→∞⁡Ext-uxt-δt,t,rtp≤λ1c1ε,which is (37).
In the case whenλ1=0, from (51), we have (53)c1Ext-uxt-δt,t,rtp≤K2eεt,which can get (39).
Now we show (40). Rewriting the integral form of (41), we have (54)eεtVxt-uxt-δt,t,rt,t,rt-Vx0-ux-δ0,0,r0,0,r0=∫0teεtεVxs-uxs-δs,s,rs,s,rs+LVxs,xs-δs,s,rsds+M2t,where M2(t) is the same as (42). Proceeding as in the proof before, with λ1=0, we can state that (55)c1eεtxt-uxt-δt,t,rtp≤K2+M2t.By the nonnegative semimartingale convergence theorem (see [26]), we have (56)limsupt→∞⁡c1eεtxt-uxt-δt,t,rtp<∞,a.s.,which means there exists a finite positive random variable η, such that (57)sup0≤t<∞eεtxt-uxt-δt,t,rtp≤η,a.s.This leads to the desired estimate (40).Remark 9.
In Theorem8, we obtain the results similar to those of Theorem 2 in [20] formally, but the delay δ(t) considered in this paper is unbounded, while the delay function δ(t) in [20] is bounded. e-εδ(t) plays an important role in dealing with problems of unboundedness. Moreover, the expressions of both 0<p≤1 and p>1 are unified in the estimation (47), which is important for the proof of the results.The following result will establishpth moment exponential stability and almost sure exponential stability of the solution x to system (2).Theorem 10.
Let Assumptions2–5 be satisfied, with λ1=0; then for any constant p>0 and initial condition (3), the unique solution x of model (2) obeys (58)limsupt→∞⁡ln⁡Extpt≤-ε,(59)limsupt→∞⁡lnxtt≤-εpa.s.,where ε∈(0,ε0), while ε0 is the unique positive root to the equation λ2=λ3+c2ε1+βp/c1(1-δ-).Proof.
By recalling (53), we know that (60)c1Ext-uxt-δt,t,rtp≤K2eεt.Letting K3=K2/c1, we have (61)eεtExt-uxt-δt,t,rtp≤K3.
Now, we need to discuss two cases when0<p≤1 and p>1.
For0<p≤1, by (9), (15), and (61), we obtain (62)eεtExtp≤eεtExt-uxt-δt,t,rtp+eεtEuxt-δt,t,rtp≤K3+eεtEuxt-δt,t,rtp≤K3+βpeεt-δtExt-δtp.From (62), for any T>0, we have (63)sup0≤t≤TeεtExtp≤K3+βpsup0≤t≤Teεt-δtExt-δtp≤K3+βpsup-δ0≤t≤0eεtExtp+βpsup0≤t≤TeεtExtp≤K3+βpk0p+βpsup0≤t≤TeεtExtp,where k0 is a upper bound of initial condition (3) as defined in the proof of Theorem 6. Noting that 1-βp>0, we have (64)sup0≤t≤TeεtExtp≤K3+βpk0p1-βp≤1+1-β1-pK3+k0p1-βp.
For the case whenp>1, letting η=β/1-β in (10), where β is defined in Assumption 5, from (61), we obtain (65)eεtExtp≤1-β1-peεtExt-uxt-δt,t,rtp+β1-peεtEuxt-δt,t,rtp≤1-β1-pK3+βeεt-δtExt-δtp.Inequality (65) follows that, for any T>0, (66)sup0≤t≤TeεtExtp≤1-β1-pK3+βsup0≤t≤Teεt-δtExt-δtp≤1-β1-pK3+βsup-δ0≤t≤0eεtExtp+βsup0≤t≤TeεtExtp≤1-β1-pK3+k0p+βsup0≤t≤TeεtExtp.From (66), we have (67)sup0≤t≤TeεtExtp≤1+1-β1-pK3+k0p1-βp.
LettingT→∞ in (64) and (67), we get, for any p>0, (68)sup0≤t≤∞eεtExtp≤1+1-β1-pK3+k0p1-βp.A routine computation to (68) gives rise to the desired estimate (58).
By (57), proceeding as in the proof of (58), we obtain that, for 0<p≤1, (69)sup0≤t≤∞eεtxtp≤η/c1+k0p1-βp,a.s.,and for p>1, (70)sup0≤t≤∞eεtxtp≤1-β1-pη/c1+k0p1-β,a.s.By (69) and (70), we have (71)limsupt→∞⁡ln⁡xtt≤-εpa.s.which is (59).Remark 11.
In Theorem10, we improve the results of Theorem 3 in [20] with the condition of delay functions from being bounded to unbounded. The proof of our results is different from that in [20] due to the unboundedness of delay functions.


## 4. Example

In this section, a numerical example is given to illustrate the effectiveness of the obtained results.To avoid the tedious computation, we consider the example whenp=2, it is appropriate to illustrate the results. As a premise, in model (2) with initial condition (3), provide that w(t) is a scalar Brownian motion; r(t) is a right-continuous Markov chain on the state space S={1,2} with the generator Γ=-112-2.Lettingδ(t)=0.5t+1, it is clear that Assumption 3 holds. Define (72)fx,y,t,1=-9x-0.2125e-0.5ty,fx,y,t,2=-8x-0.475e-0.5ty,gx,y,t,1=0.4e-0.5ty,gx,y,t,2=0.3e-0.5ty,ux,t,1=0.025e-0.5tx,ux,t,2=0.05e-0.5tx.Then the functions f and g satisfy Assumption 2, while the neutral term u:(R×R+×S)→R+ obeys Assumption 5 with β=0.2.For the sake of showing our problem well and truly, we defineV(x,t,1)=x2, V(x,t,2)=2x2. That is in support of conditions (13) with c1=1, c2=2 in Assumption 4.Under the expressions aforesaid, as well as formula (5), we compute (73)LVx,y,t,1≤-17x2+2.6×0.5e-20.5t+1y2,LVx,ty,t,2≤-17×2x2+2.6×0.5e-20.5t+12y2.Consequently, we infer the fact that, for all x,y∈R, t∈R+, and i∈S, condition (14) holds with parameters δ-=0.5, λ1=0, λ2=17, and λ3=2.6. Furthermore, we get ε0=2.5 from (38). So it could be a consideration of the case when ε=2.Based on the above argument, we can show that the example is advisable, and model (2) comes into following scalar equation: (74)dxt-0.025e-0.5tx0.5t-1=-9xt-0.2125e-0.5tx0.5t-1dt-0.4e-0.5tx0.5t-1dwt,dxt-0.05e-0.5tx0.5t-1=-8xt-0.475e-0.5tx0.5t-1dt-0.3e-0.5tx0.5t-1dwt.Applying Theorems 8 and 10, we can conclude that the solution of (74) obeys (75)limsupt→∞⁡ln⁡xt-uxt-δt,t,rtpt≤-ε=-2,a.s.,limsupt→∞⁡lnxtt≤-εp=-1a.s.,where ε=2 and p=2. Figures 1 and 2 give computer simulations of ln⁡xt/t and ln⁡xt-uxt-δt,t,rt/t for the solution of (74) using the Euler-Maruyama method (see [13]) with step size 0.01 and initial data ϕ(s)=2+s, s∈[-δ(0),0], r(0)=2 for 200 samples. Figures 1 and 2 illustrate the results accurately.Figure 1
Computer simulation ofln⁡xt-uxt-δt,t,rt/t of the solution x(t) of (74).Figure 2
Computer simulation ofln⁡xt/t of the solution x(t) of (74).Remark 12.
We use the Euler-Maruyama method which was investigated in [20] to simulate the numerical example considered above. In [20], the Euler-Maruyama method was established under both bounded delay and unbounded delay conditions. The above example we considered satisfies not only the conditions of Assumptions 2–5 in this paper, but also the conditions of A1–A7 in [20]; hence it is applicative to stimulate the example with the Euler-Maruyama method.


## 5. Conclusion

In this paper, we have improved the stability result of a class of hybrid stochastic neutral differential equations with unbounded delay. The method of Lyapunov functions ande-εδ(t) function are fully used to solve our problems. Our main contribution is generalizing the delay functions from bounded case to the unbounded in NSDE with time-dependent delay and Markovian switching. The concurrent consideration of neutral term, unbounded delay, and Markovian switching in this paper makes our results more general and adoptable to many practical systems. Compared with the existing literatures, one contribution of this paper is that our results advance the delay terms (δ(t)=θt,0<θ<1) in [21] to general unbounded functions and add the neutral term. The other contribution is that we develop the pth and almost sure exponential stability criteria under the unbounded delay conditions, which generalizes and improves the results greatly in [20] where the delay function is limited by supt≥0δ(t)=h<∞.



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*Source: 2941349-2017-08-21.xml*