# The Averaging Principle for Stochastic Pantograph Equations with Non-Lipschitz Conditions

Authors: Dongdong Gao; Jianli Li; Zhiguo Luo; Danfeng Luo
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper is devoted to presenting an averaging principle for stochastic pantograph equations. Under suitable non-Lipschitz conditions, the solutions to stochastic pantograph equations can be approximated by solutions to averaged stochastic systems in the mean-square sense and probability. At last, an example is given to demonstrate the feasibility of obtained results. Moreover, our results have generalized significantly some previous ones.


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## Body

## 1. Introduction

Pantograph equations [1] are a kind of equations with unbounded delay, and they were used in describing the various phenomena like biology, electrodynamics, economy, and some other nonlinear dynamical systems [2–4]. Based on these irreplaceable roles, the existence, uniqueness, and stability for different kinds of pantograph equations were tremendously investigated by many scholars. Of course, some excellent and important articles have also emerged in our vision (see [5–8] and references therein).On the contrary, the averaging principle as a powerful method has been largely applied in stochastic differential systems, and its main role is to strike a balance between complex models that are more realistic and simpler models that are more amenable to analysis and simulation. Referring to the averaging principle, we are indispensable to recall some excellent articles [9–13], which have discussed the corresponding solutions to stochastic differential equations by the averaging principle.To the best of authors’ knowledge, there is no paper which has considered the approximation theorem as an averaging principle for stochastic pantograph equations. To fill this gap, in this paper, we are intended to study the stochastic pantograph equations as follows:(1)dXt=ft,Xt,Xθtdt+gt,Xt,XθtdBt,X0=X0,where t∈0,T, X0∈ℝn is the initial value, which is ℱ0-measurable on ℝn and satisfying EX02<∞, 0<θ<1, f:0,T×ℝn×ℝn⟶ℝn, g:0,T×ℝn×ℝn⟶ℝn×m, and Bt is a m-dimensional Brownian motion on the complete probability space Ω,ℱ,ℙ.The highlights and major contributions of this paper are reflected in the subsequent key aspects:(i)
We first attempt to investigate the property of solutions for a class of stochastic pantograph equations by the averaging principle under the non-Lipschitz conditions. Comparing with some previous literatures [5–8], the corresponding conditions are required to satisfy the Lipschitz condition or the local Lipschitz condition. However, in some practical cases, the Lipschitz condition is usually violated. Therefore, the Lipschitz condition will be replaced by the non-Lipschitz conditions which are much weaker than the Lipschitz condition in our paper.(ii)
The previous literatures [9–11] have not considered the effect of delay terms on the averaging principle for corresponding stochastic system. However, the delay effects do exist in these stochastic differential systems. Therefore, in this paper, we consider a kind of delay stochastic differential equations with a linear delay τt=θt with 0<θ<1.The arrangement of the rest paper is as follows. In Section2, necessary definition and hypothesis conditions will be presented. Section 3 is devoted to present the averaging principle for stochastic pantograph equation (1). In Section 4, we will give an example to demonstrate the feasibility of theoretical results obtained in Section 3.


## 2. Preliminaries

In this section, we will give some necessary definition and hypothesis conditions, which will be used in later sections.Definition 1.
Anℝn-value stochastic process Xt0≤t≤T is called a unique solution of (1) if the following conditions hold:(i)
Xt is continuous with respect to t and ℱt adapted.(ii)
ft,Xt,Xθt∈ℒ10,T;ℝn and gt,Xt,Xθt∈ℒ20,T;ℝn×m.(iii)
For allt∈0,T, we have(2)Xt=X0+∫0tfs,Xs,Xθsds+∫0tgs,Xs,XθsdBs,a.s.(iv)
For any other solutionX∗t, we have PXt=X∗t,∀t∈0,T=1.
Next, we list the necessary hypothesis conditions, which will be used to prove the main results in next section.
(H1) (non-Lipschitz condition): there exists a functionΦ· such that, for any fixed t≥0 and Xi,Yi∈Rn,i=1,2, we have(3)ft,X1,Y1−ft,X2,Y22∨gt,X1,Y1−gt,X2,Y22≤Φt,X1−X22,Y1−Y22,where Φ· satisfies Φt,0,0=0 and define Ω=0,+∞×0,+∞ and ∫Ω1/Φt,u,vdudv=∞ and there exist nonnegative functions λit,i=1,2,3 such that, for u,v≥0, Φt,u,v≤λ1t+λ2tu+λ3tv and ∫0Tλitdt<∞,i=1,2,3.


## 3. Main Results

This section is devoted to present an averaging principle for stochastic pantograph equations. First, we consider the standard form of the system (1):(4)Xεt=X0+ε∫0tfs,Xεs,Xεθsds+ε∫0tgs,Xεs,XεθsdBs,where ε∈0,ε0 is a positive small parameter and ε0 is a given fixed number.To obtain the averaging principle for stochastic pantograph equations, we also need following condition: there exist measurable functionsf∗:ℝn×ℝn⟶ℝn and g∗:ℝn×ℝn⟶ℝn×m such that the following holds.(H2): for anyT1∈0,T,X,Y∈ℝn, there exist two positive bounded functions ΨiT1,i=1,2, such that(5)1T1∫0T1fs,X,Y−f∗X,Y2ds≤Ψ1T1X2+Y2,1T1∫0T1gs,X,Y−g∗X,Y2ds≤Ψ2T1X2+Y2,where limT1⟶∞ΨiT1=0.By the above preparation, we now consider the original solutionXεt converges to the solution Xε∗t of the averaged system:(6)Xε∗t=X0+ε∫0tf∗Xε∗s,Xε∗θsds+ε∫0tg∗Xε∗s,Xε∗θsdBs,as ε goes to zero and t∈0,T.Theorem 1.
Assume that conditions (H1) and (H2) hold. Then, for a given arbitrarily small numberδ1>0, there exist M>0,ε1∈0,ε0 and β∈0,1 such that, for all ε∈0,ε1, we have(7)Esupt∈0,Mε−βXεt−Xε∗t2≤δ1.Proof.
For anyt∈0,u⊂0,T, by (4) and (6), we have(8)Xεt−Xε∗t=ε∫0tfs,Xεs,Xεθs−f∗Xε∗s,Xε∗θsds+ε∫0tgs,Xεs,Xεθs−g∗Xε∗s,Xε∗θsdBs.
Taking mathematical expectation on above equation and applying the elementary inequality, we have(9)Esup0≤t≤uXεt−Xε∗t2≤2ε2Esup0≤t≤u∫0tfs,Xεs,Xεθs−f∗Xε∗s,Xε∗θsds2+2εEsup0≤t≤u∫0tgs,Xεs,Xεθs−g∗Xε∗s,Xε∗θsdBs2=I1+I2.
Using the elementary inequality again, we have(10)I1≤4ε2Esup0≤t≤u∫0tfs,Xεs,Xεθs−fs,Xε∗s,Xε∗θsds2+4ε2Esup0≤t≤u∫0tfs,Xε∗s,Xε∗θs−f∗Xε∗s,Xε∗θsds2=I11+I12.
From (H1) and Cauchy–Schwarz inequality, we have(11)I11≤4ε2uEsup0≤t≤u∫0tfs,Xεs,Xεθs−fs,Xε∗s,Xε∗θs2ds≤4ε2uEsup0≤t≤u∫0tΦs,Xεs−Xε∗s2,Xεθs−Xε∗θs2ds≤4ε2uEsup0≤t≤u∫0tλ1s+λ2sXεs−Xε∗s2+λ3sXεθs−Xε∗θs2ds≤4ε2uusup0≤t≤uλ1t+Esup0≤t≤u∫0tλ2sXεs−Xε∗s2+λ3sXεθs−Xε∗θs2ds≤4ε2u2sup0≤t≤uλ1t+4ε2u∑i=23sup0≤t≤uλit∫0uEsup0≤s1≤sXεs1−Xε∗s12ds.
Also, from (H2) and Cauchy–Schwarz inequality, we have(12)I12≤4ε2uEsup0≤t≤u∫0tfs,Xε∗s,Xε∗θs−f∗Xε∗s,Xε∗θs2ds≤4ε2u2Esup0≤t≤u1t∫0tfs,Xε∗s,Xε∗θs−f∗Xε∗s,Xε∗θs2ds≤4ε2u2sup0≤t≤uΨ1tEsup0≤t≤uXε∗t2+sup0≤t≤uXε∗θt2≤8ε2u2sup0≤t≤uΨ1tEsup0≤t≤uXε∗t2.
Analogously, by using the elementary inequality, we also have(13)I2≤4εEsup0≤t≤u∫0tgs,Xεs,Xεθs−gs,Xε∗s,Xε∗θsdBs2+4εEsup0≤t≤u∫0tgs,Xε∗s,Xε∗θs−g∗Xε∗s,Xε∗θsdBs2=I21+I22.
From (H1) and Burkholder–Davis–Gundy inequality, we have(14)I21≤16εEsup0≤t≤u∫0tgs,Xεs,Xεθs−gs,Xε∗s,Xε∗θsds2≤16εusup0≤t≤u∫0tgs,Xεs,Xεθs−gs,Xε∗s,Xε∗θs2ds≤16εuEsup0≤t≤u∫0tΦs,Xεs−Xε∗s2,Xεθs−Xε∗θs2ds≤16εuEsup0≤t≤u∫0tλ1s+λ2sXεs−Xε∗s2+λ3sXεθs−Xε∗θs2ds≤16εuusup0≤t≤uλ1t+Esup0≤t≤u∫0tλ2sXεs−Xε∗s2+λ3sXεθs−Xε∗θs2ds≤16εu2sup0≤t≤uλ1t+16εu∑i=23sup0≤t≤uλit∫0uEsup0≤s1≤sXεs1−Xε∗s12ds.
Also, from (H2) and Burkholder–Davis–Gundy inequality, we have(15)I22≤16εEsup0≤t≤u∫0tgs,Xε∗s,Xε∗θs−g∗Xε∗s,Xε∗θsds2≤16εu2Esup0≤t≤u1t∫0tgs,Xε∗s,Xε∗θs−g∗Xε∗s,Xε∗θs2ds≤16εu2sup0≤t≤uΨ2tEsup0≤t≤uXε∗t2+sup0≤t≤uXε∗θt2≤32εu2sup0≤t≤uΨ2tEsup0≤t≤uXε∗t2.
According to the estimation ofI11,I12,I21, and I22, we have(16)Esup0≤t≤uXεt−Xε∗t2≤4ε2u2sup0≤t≤uλ1t+16εu2sup0≤t≤uλ1t+8ε2u2sup0≤t≤uΨ1tEsup0≤t≤uXε∗t2+32εu2sup0≤t≤uΨ2tEsup0≤t≤uXε∗t2+4ε2u∑i=23sup0≤t≤uλit∫0uEsup0≤s1≤sXεs1−Xε∗s12ds+16εu∑i=23sup0≤t≤uλit∫0uEsup0≤s1≤sXεs1−Xε∗s12ds≤4Mλε2u2+16Mλεu2+8MΨε2u2+32MΨεu2+8ε2u+32εu∫0uEsup0≤s1≤sXεs1−Xε∗s12ds,where sup0≤t≤uΨ1tEsup0≤t≤uXε∗t2,sup0≤t≤uΨ2tEsup0≤t≤uXε∗t2=MΨ, and sup0≤t≤uλit,i=1,2,3=Mλ. With the help of the Gronwall–Bellman inequality, we get(17)Esup0≤t≤uXεt−Xε∗t2≤4Mλε2u2+16Mλεu2+8MΨε2u2+32MΨεu2e8ε2u2+32εu2,which implies that there exists a M>0 and β∈0,1 such that for any t∈0,Mε−β⊂0,T having(18)Esupt∈0,Mε−βXεt−Xε∗t2≤Kε1−β,where(19)K=4MλM2ε1−β+16MλM2ε−β+8MΨM2ε1−β+32MΨM2ε−βe8M2ε2−2β+32M2ε1−2βis a constant. As a result, for arbitrary given number δ1, there exists ε1∈0,ε0 such that, for each ε∈0,ε1 and t∈0,Mε−β, one has(20)Esupt∈0,Mε−βXεt−Xε∗t2≤δ1.
The proof is completed.
Next, we will present the properties of the uniform convergence in probability between (4) and (6).Theorem 2.
Assume that conditions (H1) and (H2) hold. Then, for a given arbitrarily small numberδ2>0, there exist M>0,ε1∈0,ε0 and β∈0,1, such that, for all ε∈0,ε1, we have(21)limε⟶0ℙsupt∈0,Mε−βXεt−Xε∗t>δ2=0.Proof.
Givenδ2>0, with the help of Chebyshev’s inequality and Theorem 1, we obtain(22)ℙsupt∈0,Mε−βXεt−Xε∗t>δ2≤1δ22Esupt∈0,Mε−βXεt−Xε∗t2≤1δ22Kε1−β⟶0,asε⟶0.
The proof is completed.Remark 1.
IfΦt,u,v=Ku+v, the non-Lipschitz conditions reduce to a global Lipschitz conditions which have been investigated in previous literatures [5–8]. So, some previous results [5–8] will be significantly improved in our paper. Also, the averaging principle is applied to study stochastic pantograph equations for the first time. In view of the previous literatures [14, 15], the authors have studied the stochastic pantograph equations driven by Lévy noise and G-Brownian motion. To the best of our knowledge, the averaging principle can also use to study stochastic pantograph equations driven by Lévy noise and G-Brownian motion with the non-Lipschitz conditions.


## 4. Example

Consider the following stochastic pantograph equation:(23)dXεt=ε2Xεcos2t−Xεsin212tdt+εdBt,where(24)Xε=X0,t∈0,π,ft,Xεt,Xεθt=2Xεcos2t−Xεsin212t,gt,Xεt,Xεθt=1.Letg∗t,Xεt,Xεθt=1, and(25)f∗t,Xεt,Xεθt=1π∫0πft,Xεt,Xεθt=Xε2.Therefore, the simplified stochastic pantograph equation (23) can be defined as(26)dXε∗t=εXε∗2dt+εdBt.It is clear to see all conditions (H1) and (H2) hold for the functions defined in systems (23) and (26). By the Theorems 1 and 2, we obtain(27)Esupt∈0,Mε−βXεt−Xε∗t2≤δ1,limε⟶0ℙsupt∈0,Mε−βXεt−Xε∗t>δ2=0.


## 5. Conclusion

In this paper, we mainly discuss an averaging principle for stochastic pantograph equations at the first time. Under suitable non-Lipschitz conditions, we obtain that the solutions to stochastic pantograph equations can be approximated by solutions to averaged stochastic systems in the mean-square sense and probability. Then, an example is given to demonstrate the feasibility of obtained results.In future work, the interesting extension of our study would be to discuss the averaging principle for the impulsive stochastic pantograph equations with time delays and the fractional stochastic pantograph equations.



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