# On Random Periodic Solution to a Neutral Stochastic Functional Differential Equation

Authors: Lili Gao; Litan Yan
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

In this paper, we consider the random periodic solution to a neutral stochastic functional differential equation driven by Brownian motion. We obtain the existence and uniqueness of the random periodic solution by Banach fixed point theorem. Moreover, we introduce two examples to illustrate our results.


---

## Body

## 1. Introduction

In the last two decades, the theory of stochastic dynamical systems has attracted much attention due to its wide applications in the fields such as financial market, insurance, biology, medical science, population dynamic, and control (see [1–8]). The idea to regard stochastic differential equations (SDEs) as random dynamical systems can be traced back to late 1970s and early 1980s (see [9–11]). The study of periodic solutions has occupied a central role in the theory of dynamical system since Poincaré’s seminal work [12]. Periodic solutions of stochastic differential equations or stochastic partial differential equations (SPDEs) have also been studied by a number of authors such as Chojnowska-Michalik [13, 14], Fife [15], Vejvoda [16], and Zhang and Zhao [17]. On the other hand, the theory of stochastic functional differential equations (SFDEs) has received a great deal of attention in the last two decades. More recently researchers have given special interest to the study of equations in which the variable delay argument occurs in the derivative of the state variable, or so-called neutral stochastic functional differential equations (NSFDEs). Many well-known theorems in SFDEs are successfully extended to NSFDEs; see, for example, [18–21]. However, contrast to the extensive studies on the periodicity of usual stochastic differential equations, there has been little systematic investigation on stochastic periodic solutions of NSFDEs. The main reason for this is the complexity of dependence structures such equations. In this paper, we consider the random periodic solutions of such stochastic equations.Random periodic behaviour is more accurate to describe the real world problems, e.g., in biological, environmental, and economic systems. But these problems are often subject to random perturbations or under the influence of noises. In our opinion, it is greatly important to study the pathwise random periodic solutions of random dynamical systems. The definition of the pathwise random periodic solutions was firstly given by Zhao and Zheng [17]. Later the definition of random periodic solutions and their existence for semi-flows generated by nonautonomous SDEs and SPDEs with additive noise were given by Feng and Zhao in [22, 23].It is well-known that the most powerful method to prove the existence of the periodic solution in the deterministic case is, for example, fixed point theorem, monotone semi-flow, topological degree theorem, and Lyapunov method (see, for example, [24–28]). But the existence of the periodic solutions of the stochastic differential systems is more complex than the deterministic case. Recently, there are some works studying existence random periodic solutions for SDEs, such as Fenget al. [29–31]. In [30], Fenget al. study the existence of random periodic solutions to semilinear stochastic differential equations. In [31], an ergodic theory of random dynamical systems has been built under the stationary regime, in which stationary solutions and stationary measures, which are “equivalent,” are fundamental objects. In [29], random periodic solutions and their approximating by the Euler-Maruyama scheme were studied.Motivated by the above, in this paper we consider the existence of random periodic solution to NSFDEs of the form(1)dxt0t-Dxtt0=Axt0tdt+ft,xtt0dt+gt,xtt0dWt,xt0t0=ξ.For the equation, we introduce some assumptions which are very natural. Let τ>0 and t0∈R be two given constants and let C([-τ,0];Rd) denote the space of continuous functions φ from [-τ,0] to Rd with the usual norm φ=sup-τ≤θ≤0φθ. Moreover, we also claim that (i)
f:[t0,T]×C([-τ,0];Rd)→Rd and g:[t0,T]×C([-τ,0];Rd)→Rd×m are Borel measurable functions and there is a constant l>0 such that (2)ft+l,ϕ=ft,ϕ,gt+l,ϕ=gt,ϕfor anyt∈R,ϕ,φ∈C([-τ,0];Rd), i.e., f and g are l-periodic in time;(ii)
D:C([-τ,0];Rd)→Rd is Borel measurable with D(0)=0;(iii)
A is a symmetric and negative-definite d×d matrix;(iv)
W={W(t),t∈R} is a two-sided Wiener process in Rm defined on a probability space (Ω,F,P). The filtration is defined as follows: (3)Fst=σWu-Wv:s≤v≤u≤t,Ft=F-∞t=⋁s≤tFst;(v)
xt={x(t+θ):-τ≤θ≤0} is regarded as a C([-τ,0];Rd)-valued stochastic process;(vi)
the initial dataξ={ξ(θ):-τ≤θ≤0} is Ft0-measurable C([-τ,0];Rd)-valued random variable such that Eξ2<∞.By the variation of constant formula, we can get the solution of (1) as follows:(4)xt0t,ξ=eAt-t0ξ0-Dξ+Dxtt0+∫t0teAt-sADxst0ds+∫t0teAt-sfs,xst0ds+∫t0teAt-sgs,xst0dWs,which is equivalent to (5)xt0t,ξ=Dxtt0+∫-∞teAt-sADxst0ds+∫-∞teAt-sfs,xst0ds+∫-∞teAt-sgs,xst0dWs. Denote the standard P-preserving ergodic Wiener shift by θ:R×Ω→Ω, (6)θtωs:Wt+s-Wt,t,s∈R, and u(t,t0)ξ=xt0(t,ξ) for all t≥t0. We then see that the solution of (1) satisfies the following semi-flow property and periodicity: (7)ut,r,ω=ut,s,ω∘us,r,ω,ut+l,s+l,ω=ut,s,θlωfor all r<s≤t.For convenience, if there is no confusion, usually initial valueξ or ω will be omitted. Let xt0(r) be the solution of (1) starting from time t0.The paper is organized as follows. In Section2, we briefly present some basic notations and preliminaries. In Section 3, we give some basic lemmas and existence of random periodic solution of (1). In Section 4, some examples are given to illustrate our result.


## 2. Preliminaries

In this section, we will give the definition of the random periodic solution and present briefly some basic notations.Definition 1 (see [22, 23]).
A random periodic path of periodl of the semi-flow u:Δ×Ω×Rd→Rd is an F-measurable map Y:R×Ω→Rd such that (8)ut,s,ωYs,ω=Yt,ω,Ys+l,ω=Ys,θlωfor any (t,s)∈Δ and almost every ω∈Ω, where Δ≔{(t,s)∈R2,s≤t}, (Ω,F,P,(θt)t∈R) is a metric dynamical system.We introduce the next notations.(i)
Let{λj,j=1,2,⋯,d} denote eigenvalues of a negative-definite d×d-symmetric matrix A such that (9)0>λ1>λ2>⋯>λd.(ii)
Forp≥1, we denote the Lp-norm of a random process ξ by (10)ξp=supt∈REξp1/p.(iii)
For ad1×d2 matrix A=(aij), we denote by |A| the Hilbert-Schmidt norm (Frobenius norm) (11)A=∑i=1d1∑j=1d2aij2.We introduce some assumptions about (1).Condition 1.
Assume that there exist two constantsK,K¯>0 such that(12)ft,φ-ft,ϕ2∨gt,φ-gt,ϕ2≤Kφ-ϕ2,(13)ft,φ2∨gt,φ2≤K¯1+φ2.
Moreover, assume also that there are the other two constantsκ∈(0,1),κ¯>0 such that(14)Dφ-Dϕ≤κφ-ϕ,(15)ADϕ2≤κ¯1+ϕ2
for anyt∈R and φ,ϕ∈C([-τ,0];Rd).Condition 2.
There exists a constantK∗>0, such that (16)ξ2≤K∗.Lemma 2 (see [32]).
If (14) holds, we then have (17)φ0-Dφp≤1+κpφp for all φ∈C([-τ,0];Rd) and p≥1.Lemma 3 (see [32]).
If (14) holds, we have (18)supt0≤s≤txsp≤κ1-κξp+11-κpsupt0≤s≤txs-Dxsp for all p>1.Lemma 4.
Denote byx(t),y(t) two solutions of system (1) with different initial values ξ,η, respectively. If (14) holds, we have (19)supt0≤s≤txs-ysp≤κ1-κξ-ηp+11-κpsupt0≤s≤txs-ys-Dxs-Dysp for all p>1.Proof.
For anyε>0, we have that (20)xs-ysp=Dxs-Dys+xs-ys-Dxs-Dys≤1+ε1/p-1p-1Dxs-Dyspε+xs-ys-Dxs-Dysp≤1+ε1/p-1p-1κpxs-yspε+xs-ys-Dxs-Dysp. Let ε=(κ/(1-κ))p-1 yield that (21)xs-ysp≤κxs-ysp+11-κp-1xs-ys-Dxs-Dysp. Therefore (22)supt0≤s≤txs-ysp≤κsupt0≤s≤txs-ysp+11-κp-1supt0≤s≤txs-ys-Dxs-Dysp≤κξ-ηp+κsupt0≤s≤txs-ysp+supt0≤s≤txs-ys-Dxs-Dysp, and the required assertion follows immediately.Lemma 5 (see [33]).
LetReλ(A)=max{Reλ:λ be an eigenvalue of A}. If Reλ(A)<0 then there exist α>0 and M>0 such that (23)eAt≤Me-αt,t≥0.


## 3. Existence and Uniqueness of Random Periodic Solution

In this section, we consider the random periodicity of (1).Lemma 6.
Let the matrixA be as stated above and let Conditions 1 and 2 hold. Then there exists a constant C>0 such that, for any k∈N,r≥t0, the solution of (1) satisfies (24)Esupt0-τ≤s≤rxt0s2≤C.Proof.
Using Itô’s formula toe2αrxt0r-Dxrt02, we have that(25)e2αrxt0r-Dxrt02=e2αt0ξ0-Dξ2+2α∫t0re2αsxt0s-Dxst02ds+2∫t0re2αsxt0s-Dxst0TAxt0sds+2∫t0re2αsxt0s-Dxst0Tfs,xst0ds+∫t0re2αsgs,xst02ds+2∫t0re2αsxt0s-Dxst0Tgs,xst0dWs≡∑j=16Ψjfor any k∈N,r≥t0. We need to estimate the six terms in the right hand side of (25).
For the first term, by Lemma2 we have (26)ξ0-Dξ2≤1+κ2ξ2.
For the third term, we have(27)Ψ3=2∫t0re2αsxt0s-Dxst0TAxt0s-Dxst0ds+2∫t0re2αsxt0s-Dxst0TADxst0ds. By Lemma 2 and (15), we derive that (28)2∫t0re2αsxt0s-Dxs-klTADxs-klds≤2∫t0re2αsxt0s-Dxst0ADxst0ds≤∫t0re2αsxt0s-Dxst02+ADxst02ds≤∫t0re2αs1+κ2xst02+κ¯1+xst02ds≤∫t0re2αs1+κ2+κ¯1+xst02ds, Notice that (29)2α∫t0re2αsxt0s-Dxst02ds+2∫t0re2αsxt0s-Dxst0TAxt0s-Dxst0ds=2∫t0re2αsxt0s-Dxst0TαI+Axt0s-Dxst0ds≤0, since the matrix αI+A is non-positive definite. We get that (30)Ψ2+Ψ3≤∫t0re2αs1+κ2+κ¯1+xst02ds. Similarly, by Lemma 2 and (13), we can show that (31)Ψ4=2∫t0re2αsxt0s-Dxst0Tfs,xst0ds≤∫t0re2αs1+κ2+K¯1+xst02ds,Ψ5=∫t0re2αsgs,xst02ds≤∫t0re2αsK¯1+xst02ds. It follows from (25) that(32)Esupt0≤s≤re2αsxt0s-Dxst02≤e2αt01+κ2Eξ2+C1∫t0re2αs1+Exst02ds+Esupt0≤s≤rΨ6, where C1=2(1+κ)2+κ¯+2K¯.
On the other hand, by the Burkholder-Davis-Gundy inequalities and (13), we derive the following:(33)Esupt0≤s≤rΨ6≤82E∫t0re4αsxt0s-Dxst02gs,xst02ds1/2≤82Esupt0≤s≤re2αsxt0s-Dxst02∫t0re2αsgs,xst02ds1/2≤12Esupt0≤s≤re2αsxt0s-Dxst02+64E∫t0re2αsgs,xst02ds≤12Esupt0≤s≤re2αsxt0s-Dxst02+64K¯∫t0re2αs1+Exst02dsfor all k∈N,r≥t0. Substituting this into (32) yields that(34)e2αt0Esupt0≤s≤rxt0s-Dxst02≤Esupt0≤s≤re2αsxt0s-Dxst02≤2e2αt01+κ2Eξ2+C2∫t0re2αs1+Exst02ds, where C2=2C1+128K¯=4(1+κ)2+2κ¯+132K¯. By Lemma 3 we have that (35)Esupt0≤s≤rxt0s2≤C3Eξ2+C2e-2αt01-κ2∫t0re2αs1+Exst02ds, where C3=κ/(1-κ)+2(1+κ)2/(1-κ)2. It follows that (36)1+Esupt0-τ≤s≤rxt0s2≤1+Eξ2+Esupt0≤s≤rxt0s2≤1+1+C3ξ2+C2e-2αt01-κ2∫t0re2αs1+Esupt0-τ≤n≤sxt0n2ds for all k∈N,r≥t0. Thus, the Gronwall inequality implies that (37)1+Esupt0-τ≤s≤rxt0s2≤1+1+C3K∗expC2e2αr-t0-12α1-κ2 by Condition 2 and the lemma follows.Theorem 7.
Assume Condition1 holds, and(38)4κ2+4M2κ2A2+K2α2+2M2K2α<1,and then problem (1) has a unique random periodic solution.Proof.
From Lemma6, we have known that x(t) is bounded in L2. Define the operator L on L2(R;L2(Ω,Rd)) as follows: (39)Lxt0t≔Dxtt0+∫-∞teAt-sADxst0ds+∫-∞teAt-sfs,xst0ds+∫-∞teAt-sgs,xst0dWs. We will prove that L is a contraction. Assume that xt0(t) and yt0(t) are two solutions of (1), and xt0(t)≠yt0(t), (40)ELxt0t-Lyt0t2=EDxtt0-Dytt0+∫-∞tAeAt-sDxst0-Dyst0ds+∫-∞teAt-sfs,xst0-fs,yst0ds+∫-∞teAt-sgs,xst0-gs,yst0dWs2≤4EDxtt0-Dytt02+4E∫-∞teAt-sADxst0-ADyst0ds2+4E∫-∞teAt-sfs,xst0-fs,yst0ds2+4E∫-∞teAt-sgs,xst0-gs,yst0dWs2≤4κ2Extt0-ytt02+4∫-∞tMe-αt-sds·∫-∞tκ2Me-αt-sA2xst0-yst02ds+4∫-∞tMe-αt-sds·∫-∞tMe-αt-sK2Exst0-yst02ds+4∫-∞tM2e-2αt-sK2Exst0-yst02ds≤4κ2+4κ2M2A2α2+4M2K2α2+2M2K2αsupt∈RExt0t-yt0t2,for all t≥t0. It follows that (41)Lxt0t-Lyt0t2≤4κ2+4M2κ2A2+K2α2+2M2K2αxt0t-yt0t2 for each t∈R, which means that L is a contraction if (38) hold. By the contraction principle, there exists a unique fixed point xt0(t) for L in L2(R;L2(Ω,Rd)), such that Lxt0=xt0, that is, (1) has a unique solution.
We now show the random periodicity ofxt0(t). Notice that (42)xt0r+l=Dxr+lt0+∫-∞r+leAr+l-sADxst0ds+∫-∞r+leAr+l-sfs,xst0ds+∫-∞r+leAr+l-sgs,xst0dWs=Dxr+lt0+∫-∞reAr-sADxs+l-klds+∫-∞reAr-sfs+l,xs+lt0ds+∫-∞reAr-sgs+l,xs+lt0dW~s=Dxr+lt0+∫-∞reAr-sADxs+lt0ds+∫-∞reAr-sfs,xs+lt0ds+∫-∞reAr-sgs,xs+lt0dW~sby the continuity of f and g, where W~(s)≔(θlω)(s)=W(s+l)-W(s). On the other hand, we have (43)θlxt0r=Dθlxrt0+∫-∞reAr-sADθlxs+l-k-1lds+∫-∞reAr-sfs,θlxst0ds+∫-∞reAr-sgs,θlxst0dW~s for any r,k≥0. It follows from pathwise uniqueness of the solution of (1) that(44)xt0r,θlω=xt0r+l,ωfor any r,k≥0 and almost every ω∈Ω. That is to say, the unique solution is random periodic solution.The following result shows that the solution of (1) is continuously dependent on initial value.Theorem 8.
Denotext0(r),yt0(r) two solutions of system (1) with different initial values ξ,η, respectively, starting with the time t0. Assume that Conditions 1 and 2 hold for both initial values; we then have(45)xt0r-yt0r2≤C¯eC~Eξ-η2for any k∈N,r≥t0, where C¯=(2κ2+3κ+3)/(1-κ)2 and C~=(4(1+κ)2+2|A|2κ2+4K+128K¯)(e2α(r-t0)-1)/2α(1-κ)2.Proof.
Letxt0(r) and yt0(r) denote the solutions of system (1) starting from t0 with initial values ξ and η, respectively. According to (4), one has (46)xt0r-yt0r=eAr-t0ξ0-Dξ-eAr-t0η0-Dη+Dxrt0-Dyrt0+eAr∫t0re-AsADxst0-Dyst0ds+eAr∫t0re-Asfs,xst0-fs,yst0ds+eAr∫t0re-Asgs,xst0-gs,yst0dWs.for any k∈N,r≥t0. Denote ζt0(r)=xt0(r)-yt0(r). According to the method used in Lemma 6, we get the following result similar to (32):(47)Esupt0≤s≤re2αsζt0s-Dxst0-Dyst02≤e-2αkl1+κ2Eξ-η2+∫t0re2αsC1¯Exst0-yst02ds+2Esupt0≤s≤r∫t0re2αsζt0s-Dxst0-Dyst0Tgs,xst0-gs,yst0dWs by Lemma 4, where C1¯=2(1+κ)2+|A|2κ2+2K. On the other hand, by the Burkholder-Davis-Gundy inequalities and the conditions in Section 2, similar to the proof of (33), we see that (48)2Esupt0≤s≤r∫t0re2αsζt0s-Dxst0-Dyst0Tgs,xst0-gs,yst0dWs≤12Esupt0≤s≤re2αsζt0s-Dxst0-Dyst02+64K¯∫t0re2αsExst0-yst02ds for all k∈N,r≥t0. Combining this with (47), we get (49)e2αt0Esupt0≤s≤rζt0s-Dxst0-Dyst02≤Esupt0≤s≤re2αsζt0s-Dxst0-Dyst02≤2e2αt01+κ2Eξ-η2+2C1¯+128K¯∫t0re2αsExst0-yst02ds for all k∈N,r≥t0. It follows that(50)Esupt0≤s≤rζt0s-Dxst0-Dyst02≤21+κ2Eξ-η2+C2¯e-2αt0∫t0re2αsExst0-yst02ds, where C2¯=2C1¯+128K¯. Applying Lemma 5, we see that (51)Esupt0≤s≤rζt0s2≤C3¯Eξ-η2+C2¯e-2αt01-κ2∫t0re2αsExst0-yst02ds, where C3¯=κ/(1-κ)+2(1+κ)2/(1-κ)2=(κ2+5κ+2)/(1-κ)2. Consequently (52)Esupt0-τ≤s≤rζt0s2≤Eξ-η2+Esupt0≤s≤rζt0s2≤1+C3¯Eξ-η2+C2¯e-2αt01-κ2∫t0re2αsExst0-yst02ds≤1+C3¯Eξ-η2+C2¯e-2αt01-κ2∫t0re2αsEsupt0-τ≤n≤sxt0n-yt0n2ds.for all k∈N,r≥t0. Finally, by the Gronwall inequality, we obtain that (53)Esupt0-τ≤s≤rζt0s2≤1+C3¯Eξ-η2exp∫t0rC2¯e-2αt01-κ2e2αsds=C¯eC2¯e2αr-t0-1/2α1-κ2Eξ-η2, and the required assertion (45) follows by taking C¯=(2κ2+3κ+3)/(1-κ)2 and C2¯=4(1+κ)2+2|A|2κ2+4K+128K¯.


## 4. Examples

By using the Euler-Maruyama method for NSFDEs established in [34], we can simulate the solution of (1).Example 1.
Consider the NSDE (1) with (54)Dxt=0.5xt-1,A=-5,ft,xt=xt-1,gt,xt=0.1.We simulate two samples as shown in Figures1 and 2. We can still see some periodic property of the solutions in these two figures though they are not really periodic.Figure 1
Random periodic solution with small noise: sample one.Figure 2
Random periodic solution with small noise: sample two.Example 2.
The noise in the above example is relevantly small. We consider relatively larger noise withA,f and g as follows: (55)Dxt=0.5xt-1,A=-5,ft,xt=xt-1,gt,xt=1.We simulate two different samples as demonstrated in Figures3 and 4. We can also see periodicity in Figures 3 and 4.Figure 3
Random periodic solution with large noise: sample oneFigure 4
Random periodic solution with large noise: sample two.



---

*Source: 8353065-2018-11-04.xml*