# Periodic Averaging Principle for Neutral Stochastic Delay Differential Equations with Impulses

Authors: Peiguang Wang; Yan Xu
Journal: Complexity
 (2020)

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## Abstract

In this paper, we study the periodic averaging principle for neutral stochastic delay differential equations with impulses under non-Lipschitz condition. By using the linear operator theory, we deal with the difficulty brought by delay term of the neutral system and obtain the conclusion that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. At last, an example is presented to show the validity of the proposed theories.


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## Body

## 1. Introduction

Delay, impulse, and noise are natural phenomena in most practical issues and those phenomena are generally modeled by stochastic delay differential equations with impulses, and noise can be described by Brownian motion. Recent theoretical and computational advancements indicate that stochastic delay differential equations with impulses tend to generate rich and complex dynamics. However, because of the complexity of the system, it is difficult to obtain the exact solution of the vast majority of stochastic delay differential equations with impulses. In this background, it is very important to look for an approximate system which is more amenable for analysis and simulation, and it governs the evolution of the original system over a long time scale.The averaging principle is an effective method to understand the main part of the behavior of dynamical systems. It allows to avoid the detailed analysis of complex original systems and consider the simplified equations. The first analysis for averaging principle for stochastic differential equations was deeply addressed by Khasminskij [1]. And then the averaging principle has been applied for various types of stochastic differential equations. Generally speaking, the results of averaging principle in standard form mainly fall into two categories. The first one is considered in slow-fast systems or two-time-scale systems. It approximates coupled slow component equation in two-time-scale systems by a noncoupled equation often called an averaged equation. Many important results for averaging principle for two-time-scale stochastic differential equations have been carried out, see, for example, [2–6]. The second is approximating a nonautonomous stochastic differential equation by an autonomous stochastic differential equation. The results obtained by this method can be referred to the papers [7–9] and the references therein. However, there are few results on average principle for neutral stochastic delay differential equations. Recently, averaging principle for stochastic delay differential equations of the neutral type driven by G-Brownian motion was studied [10], in which the impulses are not considered in the system.Motivated by the previous discussion, in this paper, we study the average principle for the neutral stochastic delay differential equations with impulses. We overcome the difficulties caused by the delay term which is included under the differentiation at the left-hand side, and the impulse appears in neutral stochastic differential equations.The structure of the paper is the following. In Section2, we introduce some basic concepts, notations, and necessary hypotheses. In Section 3, under several sufficient conditions, we obtain the main results that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. In Section 4, we offer an example to illustrate the effectiveness of the obtained results.


## 2. Model Description and Preliminaries

In this section, we will introduce the basic concepts, the model, and some preliminary lemmas.LetΩ,ℱ,ℱtt≥0,P be a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all P-null sets) and Bt is m-dimensional Brownian motion defined on the space. K denotes the family of all concave continuous nondecreasing functions α:ℝ+⟶ℝ+ such that α0=0,∫0+ds/αs=∞. Let C−∞,0;ℝd denote the family of all continuous functions ξ:−∞,0⟶ℝd with the norm ξ=sup−∞<θ≤0ξθ and ⋅ denote any norm in ℝd. Let Cℱt2−∞,0;ℝd denote the family of all ℱt-measurable, C−∞,0;ℝd-valued random variables ξ=ξθ:−∞<θ≤0 such that Eξ2<∞.In this paper, we will discuss the following neutral stochastic delay differential equations with impulses:(1)dxt+ctxt−τ=ft,xt,xt−τdt+gt,xt,xt−τdBt,t≠tj,j=1,2,…,l,xtj+−xtj−=Ijxtj,t=tj,j=1,2,…,l,where c∈C−∞,U,0,1 is a T-periodic function, 0<U<∞. The mappings f:0,U×ℝd×ℝd⟶ℝd and g:0,U×ℝd×ℝd⟶ℝd×m are Borel measurable and T-periodic in the first argument. Ij:ℝd⟶ℝd, xtj+ and xtj− represent the right and the left limits of xtj,j∈1,2,…,l. We assume that there exists a positive constant l such that tj+l=tj+T,Ij+lxtj+l=Ijxtj,j∈ℤ+, and 0,T∩tj,j∈ℤ+=t1,t2,…,tl. The initial condition x0 is defined by(2)x0=ξ=ξt:−∞<t≤0∈Cℱt2−∞,0;ℝd.The following lemmas are important to obtain our results.Lemma 1 (see [11]).
SupposeH is a bounded linear operator on Banach space X; if H<1, then I−H has bounded inverse operator I−H−1 and I−H−1≤1/1−H.Lemma 2 (see [11]).
SupposeH is a bounded linear operator on Banach space X and has an inverse bounded operator, for arbitrary ΔH:X⟶X, if ΔH<1/H−1, then S=H+ΔH has a bounded inverse and S−1=∑j=0∞−1jH−1H−1ΔHj.Let(3)PC−∞,U=x:xtj,tj+1−∞,U∈C−∞,0;ℝd,xtj+=xtj,j=1,2,…,l,and consider the Banach space(4)W=x:x∈PC−∞,U,xt=xt+T.LetΦ:W⟶W be defined by(5)Φxt=xt+ctxt−τ.Lemma 3 (see [12]).
LetcM=maxt∈−∞,Uct. If 0≤ct<1, then Φ has a bounded inverse Φ−1 on W, and for all x∈W,(6)Φ−1xt=∑j≥0∏0≤i≤j−1−1jct−iτxt−jτand Φ−1x≤x/1−cM.SinceΦ and Φ−1 are both linear operators, we can change system (1) by using the inverse transformation of Φ into the following form:(7)dut=ft,Φ−1ut,Φ−1ut−τdt+gt,Φ−1ut,Φ−1ut−τdBt,t≠tj,j=1,2,…,l,utj+−utj−=ΦIjΦ−1utj,t=tj,j=1,2,…,l,u0=Φ−1x0=u0.Hence,ut is an T-periodic solution of system (2) if and only if Φ−1ut is an T-periodic solution of system (1).To study the averaging principle of system (1), we impose the following hypotheses on the coefficients.H1 For all x1,y1,x2,y2∈ℝd and t∈0,U, there exist α∈K such that(8)ft,x1,y1−ft,x2,y22∨gt,x1,y1−gt,x2,y22≤αx1−y12+x2−y22,where a∨b≔maxa,b.Furthermore, by the definition ofα, there must exist positive constants k1 and k2 such that(9)αr≤k1r+k2,for allr≥0.H2 For every x∈ℝd, there exists a positive constant N1 such that Ijx≤N1.Consider the standard form of system (2)(10)du∈t=εft,Φ−1uεt,Φ−1uεt−τdt+εgt,Φ−1uεt,Φ−1uεt−τdBt,t≠tj,j=1,2,…,l,uεtj+−uεtj−=εΦIjΦ−1uεtj,t=tj,j=1,2,…,l,uε0=u0.Accordingly, the standard form of system (1) is(11)dxεt+ctxεt−τ=εft,xεt,xεt−τdt+εgt,xεt,xεt−τdBt,t≠tj,j=1,2,…,l,xεtj+−xεtj−=εIjxεtj,t=tj,j=1,2,…,l,xε0=x0,where the functions f,g,Ij,j∈1,2,…,l have the same conditions as in H1andH2, and ε∈0,ε0 is a positive small parameter with ε0 is a fixed number.Letf¯x,y:ℝd×ℝd⟶ℝd, g¯x,y:ℝd×ℝd⟶ℝd×m, and ΦIΦ−1¯x:ℝd⟶ℝd be measurable functions and satisfy the conditions in H1−H2 and the following definitions:(12)f¯x,y=1T∫0Tfs,x,yds,g¯x,y=1T∫0Tgs,x,yds,ΦIΦ−1¯x=1T∑j=1lΦIjΦ−1x.We also assume that the following hypothesis is satisfied.H3 There exists a constant M>0 such that ft,x,y2<M, gt,x,y2<M, f¯x,y2<M, and g¯x,y2<M for every t∈0,U.Now, we consider the following averaged stochastic delay differential equations which correspond to the original standard form (10):(13)dvεt=εf¯Φ−1vεt,Φ−1vεt−τ+ΦIΦ−1¯vεtdt+εg¯Φ−1vεt,Φ−1vεt−τdBt,vε0=u0.LetΦ−1vεt=yεt, for system (11), we consider averaged stochastic delay differential equations:(14)dyεt=εf¯Φ−1yεt,Φ−1yεt−τ+ΦIΦ−1¯yεtdt+εg¯Φ−1yεt,Φ−1yεt−τdBt,yε0=x0.Obviously, under hypothesesH1−H2, one can follow [13, 14] to prove the existence and uniqueness of the periodic probability solutions on −∞,U of the standard form (10) and (11) and the averaged form (13) and (14), respectively. Now, we offer the proof for the relationship between xεt and yεt.


## 3. Main Results and Proofs

In this section, we will use the periodic averaging principle to investigate the neutral stochastic delay differential equations with impulses.In the rest of the paper,CiandOi,i=1,2,3, and Pj,j=1,2,3,4 are all constants. Our main results are the following.Theorem 1.
Suppose hypothesesH1−H3 are satisfied and systems (10) and (13)–(11) and (14) have solutions uε, vε, xε, and yε, respectively, where ε∈0,ε0 is a positive small parameter with ε0 a constant, then there exist constants ε1∈0,ε0, μ1>0, and μ2>0 such that, for any ε∈0,ε1,t∈0,μ2ε−1,(15)Esupt∈0,μ2ε−1uεt−vεt2≤μ1ε,and then, we obtain(16)Esupt∈0,μ2ε−1xεt−yεt2≤Φ−12μ1ε.Proof.
By using the elementary inequality, for anyt∈0,U, we have(17)uεt−vεt2≤3ε2∫0tfs,Φ−1uεs,Φ−1uεs−τ−f¯Φ−1vεs,Φ−1vεs−τds2+3ε∫0tgs,Φ−1uεs,Φ−1uεs−τ−g¯Φ−1vεs,Φ−1vεs−τdBs2+3ε2∑j=1∞ΦIjΦ−1uεtj−∫0tΦIΦ−1¯vεtds2≔J1+J2+J3.
For the first termJ1, thanks again to the elementary inequality yields:(18)Esup0≤s≤tJ1≤6ε2Esup0≤s≤t∫0sfθ,Φ−1uεθ,Φ−1uεθ−τ−fθ,Φ−1vεθ,Φ−1vεθ−τdθ2+6ε2Esup0≤s≤t∫0sfθ,Φ−1vεθ,Φ−1vεθ−τ−f¯Φ−1vεθ,Φ−1vεθ−τdθ2.
According Ho¨lder’s inequality and hypothesis H1, we arrive at(19)6ε2Esup0≤s≤t∫0sfθ,Φ−1uεθ,Φ−1uεθ−τ−fθ,Φ−1vεθ,Φ−1vεθ−τdθ2≤6ε2tk1∫0tEsup0≤θ≤sΦ−1uεθ−Φ−1vεθ2+Φ−1uεθ−τ−Φ−1vεθ−τ2ds+6ε2t2k2≤6ε2tk1∫0tΦ−12Esup0≤θ≤suεθ−vεθ2+uεθ−τ−vεθ−τ2ds+6ε2t2k2≤12ε2tk1Φ−12∫0tEsup0≤θ≤suεθ−vεθ2ds+6ε2t2k2.
Letn be the largest integer such that nT≤t. Then, for every i=1,…,n, we obtain(20)6ε2Esup0≤s≤t∫0sfθ,Φ−1vεθ,Φ−1vεθ−τ−f¯Φ−1vεθ,Φ−1vεθ−τdθ2≤12ε2Esup0≤s≤t∑i=1n∫i−1TiTfθ,Φ−1vεθ,Φ−1vεθ−τ−f¯Φ−1vεθ,Φ−1vεθ−τdθ2+12ε2Esup0≤s≤t∫nTsfθ,Φ−1vεθ,Φ−1vεθ−τ−f¯Φ−1vεθ,Φ−1vεθ−τdθ2≤72ε2nT∫0TEsup0≤θ≤sαΦ−1vεθ−Φ−1vεiT2+Φ−1vεθ−τ−Φ−1vεiT−τ2ds+36ε2Esup0≤s≤t∑i=1n∫i−1TiTfθ,Φ−1vεiT,Φ−1vεiT−τ−f¯Φ−1vεiT,Φ−1vεiT−τdθ2+12ε2Esup0≤s≤t∫nTsfθ,Φ−1vεθ,Φ−1vεθ−τ−f¯Φ−1vεθ,Φ−1vεθ−τdθ2.
By the Ho¨lder’s inequality, Burkholder–Davis–Gundy’s inequality, and hypotheses H1 and H3, it follows that(21)Esup0≤s≤tΦ−1vεs−Φ−1vεiT2+Esup0≤s≤tΦ−1vεs−τ−Φ−1vεiT−τ2≤Φ−12Esup0≤s≤tvεiT−vεiT−τ2+2Φ−12Esup0≤s≤tvεs−vεiT2≤C1+C2ε2Esup0≤s≤t∫iTsf¯Φ−1vεθ,Φ−1vεθ−τ2dθ+C2εEsup0≤s≤t∫iTsg¯Φ−1vεθ,Φ−1vεθ−τ2dθ≤C1+C2ε2MT+C2εMT,t∈i−1T,iT.
The definition off¯ implies that(22)Esup0≤s≤t∑i=1n∫i−1TiTfθ,Φ−1vεiT,Φ−1vεiT−τ−f¯Φ−1vεiT,Φ−1vεiT−τdθ2≤n∑i=1nEsup0≤s≤t∫0Tfθ,Φ−1vεiT,Φ−1vεiT−τdθ−Tf¯Φ−1vεiT,Φ−1vεiT−τ2=0.
Thus, we have(23)Esup0≤s≤tJ1≤12ε2tk1Φ−12∫0tEsup0≤θ≤suεθ−vεθ2ds+6ε2t2k2+72ε2nT∫0TEsup0≤θ≤sk1Φ−1vεθ−Φ−1vεiT2+Φ−1vεθ−τ−Φ−1vεiT−τ2+k2ds+48ε2tMT≤12ε2tk1Φ−12∫0tEsup0≤θ≤suεθ−vεθ2ds+6ε2t2k2+72ε2nk1T2C1+C2ε2MT+C2εMT+72ε2nk2T2+48ε2tMT≔εO1∫0tEsup0≤θ≤suεθ−vεθ2ds+εP1.
For the second termJ2, apply Burkholder–Davis–Gundy’s inequality to deduce(24)Esup0≤s≤tJ2≤3εC3Esup0≤s≤t∫0sgθ,Φ−1uεθ,Φ−1uεθ−τ−g¯Φ−1vεθ,Φ−1vεθ−τ2dθ≤6εC3Esup0≤s≤t∫0sgθ,Φ−1uεθ,Φ−1uεθ−τ−gθ,Φ−1vεθ,Φ−1vεθ−τ2dθ+6εC3Esup0≤s≤t∫0sgθ,Φ−1vεθ,Φ−1vεθ−τ−g¯Φ−1vεθ,Φ−1vεθ−τ2dθ.
Furthermore, on account of hypothesisH1, we have(25)Esup0≤s≤t∫0sgθ,Φ−1vεθ,Φ−1vεθ−τ−g¯Φ−1vεθ,Φ−1vεθ−τ2dθ≤Esup0≤s≤t∑i=1n∫i−1TiTgθ,Φ−1vεθ,Φ−1vεθ−τ−g¯Φ−1vεθ,Φ−1vεθ−τ2dθ+Esup0≤s≤t∫nTsgθ,Φ−1vεθ,Φ−1vεθ−τ−g¯Φ−1vεθ,Φ−1vεθ−τ2dθ≤3Esup0≤s≤t∑i=1n∫i−1TiTgθ,Φ−1vεθ,Φ−1vεθ−τ−gθ,Φ−1vεiT,Φ−1vεiT−τ2dθ+3Esup0≤s≤t∑i=1n∫i−1TiTgθ,Φ−1vεiT,Φ−1vεiT−τ−g¯Φ−1vεiT,Φ−1vεiT−τ2dθ+3Esup0≤s≤t∑i=1n∫i−1TiTg¯Φ−1vεiT,Φ−1vεiT−τ−g¯Φ−1vεθ,Φ−1vεθ−τ2dθ+4MT≤3n∫0TEsup0≤θ≤sαΦ−1vεθ−Φ−1vεiT2+Φ−1vεθ−τ−Φ−1vεiT−τ2ds+12nMT+4MT.
Hence, we obtain(26)Esup0≤s≤tJ2≤12εC3k1Φ−12∫0tEsup0≤θ≤suεθ−vεθ2ds+6εC3k2t+18εC3nk2T+18εC3nk1TC1+ε2C2MT+εC2MT+72εC3nMT+24εC3MT≔εO2∫0tEsup0≤θ≤suεθ−vεθ2ds+εP2.
For the third termJ3, utilizing hypothesis H2, we deduce(27)Esup0≤s≤tJ3≤6ε2ln+1Esup0≤s≤t∑j=1lΦIjΦ−1uεtj2+6ε21T2ltEsup0≤s≤t∑j=1l∫0sΦIjΦ−1vεθ2dθ≤6ε2l2n+1N1+6ε2l2n+12N1≔εP3.
Combining (23), (23), and (27), we conclude that(28)Esup0≤s≤tuεs−vεs2≤εO1+O2∫0tEsup0≤θ≤suεθ−vεθ2ds+εP1+P2+P3≔εO3∫0tEsup0≤θ≤suεθ−vεθ2ds+εP4,and in addition, using Gronwall’s inequality, we have(29)Esup0≤s≤tuεs−vεs2≤εP4eεO3t.
Therefore, selectingμ2>0 such that, for every t∈0,μ2ε−1⊆0,U, setting μ1=P4eO3μ2, we can choose ε1∈0,ε0 such that, for each ε∈0,ε1 and t∈0,μ2ε−1,(30)Esupt∈0,μ2ε−1uεt−vεt2≤μ1ε,and then, we obtain(31)Esupt∈0,μ2ε−1xεt−yεt2=Esupt∈0,μ2ε−1Φ−1uεt−Φ−1vεt2≤Φ−12μ1ε.Corollary 1.
Suppose hypothesesH1−H3 are satisfied. Then, for any parameter γ>0, we have(32)limε⟶0Pxεt−yεt>γ=0.Proof.
In fact, under the consequence of Theorem1, by using Chebyshev–Markov inequality, for any given number γ>0, we can derive(33)Pxεt−yεt>γ≤1γ2Exεt−yεt2≤Φ−12μ1εγ2.
The conclusion follows by lettingε⟶0.
The proof is complete.


## 4. Illustrative Example

Example 1.
Consider the following neutral stochastic delay differential equations with impulses:(34)dxεt+14xεt−τ=εxεt+xεt−τdt+2εcos2txεt−τdBt,t≠tj,j=1,2,…,l,xεtj+−xεtj−=−0.04xεtj,t=tj,j=1,2,…,l,where xεt=ξt,t∈−∞,0.
LetΦxt=xt+1/4xt−τ=ut and consider the following stochastic delay differential equations with impulses:(35)duεt=ε∑i≥0−14iuεt−iτ+∑i≥0−14iuεt−i+1τdt+2εcos2t∑i≥0−14iuεt−i+1τdBt,t≠tj,j=1,2,…,l,uεtj+−uεtj−=−0.04uεtj,t=tj,j=1,2,…,l,ft,Φ−1uεt,Φ−1uεt−τ=∑i≥0−14iuεt−iτ+∑i≥0−14iuεt−i+1τ,gt,Φ−1uεt,Φ−1uεt−τ=2cos2t∑i≥0−14iuεt−i+1τ,ΦIjΦ−1uεtj=−0.04uεtj.
Then,(36)f¯Φ−1vεt,Φ−1vεt−τ=1π∫0πft,Φ−1vεt,Φ−1vεt−τdt=∑i≥0−14ivεt−iτ+∑i≥0−14ivεt−i+1τ,g¯Φ−1vεt,Φ−1vεt−τ=1π∫0πgt,Φ−1vεt,Φ−1vεt−τdt=∑i≥0−14ivεt−i+1τ,ΦIΦ−1¯vεt=1π∑j=1qΦIjΦ−1vεtj,and define a new averaged stochastic delay differential equations:(37)dvεt=εf¯Φ−1vεt,Φ−1vεt−τ+ΦIΦ−1¯vεtdt+εg¯Φ−1vεt,Φ−1vεt−τdBt,namely,(38)dvεt=ε∑i≥0−14ivεt−iτ+∑i≥0−14ivεt−i+1τ+1π∑j=1q−0.04vεtjdt+ε∑i≥0−14ivεt−i+1τdBt.
LetΦ−1vεt=yεt, we have(39)dyεt=ε∑i≥0−14iyεt−iτ+∑i≥0−14iyεt−i+1τ+1π∑j=1q−0.04yεtjdt+ε∑i≥0−14iyεt−i+1τdBt.
Whent∈0,τ, the solution of system (3) can be deduced as(40)yεt=etyε0+ε∫0te−s∑i≥1−14iyεs−iτds+ε∫0te−s∑i≥0−14iyεs−i+1τds+εe−t1π∑j=1q−0.04yεtj+ε∫0te−s∑i≥0−14iyεs−i+1τdBs.
Whent∈τ,2τ, the solution of system (3) is deduced as(41)yεt=et−τyετ+ε∫τte−s−τ∑i≥1−14iyεs−iτds+ε∫τte−s−τ∑i≥0−14iyεs−i+1τds+εe−t−τ1π∑j=1q−0.04yεtj+ε∫τte−s−τ∑i≥0−14iyεs−i+1τdBs.
Repeat the steps above on2τ,3τ,3τ,4τ, etc., and we can get the solution yεt on the whole interval yεt. In addition, it is easy to verify that the hypotheses of Theorem 1 and Corollary 1 holds; hence, the solution of the averaged system (3) converges to that of the standard system (1) in the sense of mean square and in probability.


## 5. Conclusion

In this paper, we generalize the periodic averaging principle for neutral stochastic delay differential equations with impulses. The main difficulties in the application of periodic averaging principle for neutral system are caused by the delay term which is included under the differentiation at the left-hand side of the system. Using the linear operator theory, Theorem1 and Corollary 1 show that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. We remark that when the noises are Le´vy processes other than Brownian motion considered in this paper, and the approach presented in the paper remains valid.



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