# Multivalued Impulsive SDEs Driven by G-Brownian Noise: Periodic Averaging Result

Authors: Mahmoud Abouagwa; Anas D. Khalaf; Nadia Gul; Sultan Alyobi; Al-Sharef Mohamed
Journal: Complexity
 (2022)

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## Abstract

This paper aims to study two approximation theorems in view of the periodic averaging results for non-Lipschitz multivalued stochastic differential equations with impulses and G-Brownian motion (MISDEGs). By adopting G-Itô’s formula and non-Lipschitz condition, the solutions to the simplified MSDEGs without impulses may replace those of the initial MISDEGs in view of approximation inL2-sense and capacity. Finally, we bring a couple of two examples to enhance our theoretical results.


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## Body

## 1. Introduction

The theory of averaging principles provides a useful account of how to simplify the complex systems to be more amenable in numerical calculations and analysis. Recently, a considerable amount of literature has emerged around the theme of approximation theorems for stochastic differential equations (SDEs) [1–8]. As one of the most significant models, recently, multivalued stochastic differential equations (MSDEs) received considerable critical attention. In [9, 10], averaging principles are established for MSDEs with Gaussian noise. Guo and Pei in [11] extended the technique proposed by the authors of [10] to MSDEs fluctuating with the Poisson point process. Meanwhile, by Bihari’s inequality, Mao et al. [12] proved that the solutions of the initial non-Lipschitz MSDEs perturbed by Poisson jumps can be replaced by those of simplified MSDEs both in the probability and mean square.Impulses are of pressing need for the theory of SDEs due to their contributions in modelling processes with rapid changes at certain moments of time. Nowadays, there has been a surge of interest concerning existence, uniqueness, and stability for SDEs with impulses (ISDEs) [13–16]. Although more research has been carried out on averaging principles for MSDEs, no controlled studies have been reported for MISDEs. Basically, the idea behind the periodic averaging method for ISDEs is to allow a simplified autonomous SDE without impulses to replace original nonautonomous ISDEs [17–19].The theory ofG-expectation is at the heart of our understanding of uncertainty problems, risk measures, and optimization problems [20, 21]. Peng [20] constructed the cornerstone of G-expectation theory with its related random calculus, and SDEs perturbed with G-Brownian motion (SDEGs) became the subject of much systematic investigation [22–33]. In 2017, Ren et al. [34] studied a new model of multivalued SDEGs and satisfied the existence and uniqueness problem by means of the penalized method as well as its related stochastic control problem. However, the periodic averaging method for SDEGs and MSDEGs is rarely considered, and the only paper dealing with this issue is the one mentioned in [35].Based on the above, with the aid of the G-Itô formula andG-stochastic calculus, we, in this work, present the periodic averaging principle for MSDEGs with impulses (MISDEGs) under the Taniguchi non-Lipschitz condition [36]. This article’s contributions are highlighted as follows:(i)
The model uncertainty described by G-Brownian motion fluctuation and jumps presented by impulses show our MISDEG model’s generality.(ii)
It is observed that the proofs of Theorems 2.1, 3.1, and 3.9 in [10–12] [35], respectively, depend on the second moment boundedness property of the solution. However, in our case, Theorem 2 does not depend on the boundedness property of the second moment for MISDEG solutions. Moreover, the multivalued term in our model is of a subdifferential term, which is different from the multivalued term in [9–12].(iii)
Our non-Lipschitz condition is more general than the one used in  [9–12, 35] and considers them as special cases. Therefore, the results in [9–12, 35] are generalized and extended.Section 2 is concerned with some preliminary notions, definitions, lemmas, and interpretation of the MISDEG model. In Section 3, we give the approximation in capacity andL2-sense between the initial MISDEG and simplified MSDEGs without impulses as well as the approximation order to (1) in a bounded interval of time. Finally, we bring a couple of examples to enhance our theoretical results in Section 4.


## 2. Preliminaries

Here, we mention some notions and facts on random calculus with respect to G-Brownian motion and prepare our model.


### 2.1. Notations

In this section, we first give the notion of sublinear expectation spaceΩ,ℋ,E, where Ω is a given state set and ℋ is a linear space of real valued functions defined on Ω. The space ℋ can be considered the space of random variables.AssumeΩ be the space of all continuous ℝn-valued functions ωtt∈ℝ+, with ω0=0, equipped with the distance(1)ρω1,ω2=∑i=1∞2−imaxt∈0,iωt1−ωt2⋅∧1.Then,Ω,ρ is a metric space.For allω∈Ω, we define the canonical process Wtω=ωt, t∈0,∞. The filtration generated by the canonical process Wtt≥0 is defined as ℱt=σWs,0≤s≤t. Let(2)LlipℱT≔ψWt1,Wt2−Wt1,…,Wtn−Wtn−1:0≤t1<⋯<tn≤T,ψ∈ℂb.lipℝn×n,where n≥1 and ℂb.lipℝn×ℝn refers to the space of Lipschitz-bounded functions on ℝn×ℝn and Llipℱ=∪i=1∞Llipℱi.For anyϑω=ψWt1,Wt2−Wt1,…,Wtn−Wtn−1∈Llipℱ with 0<t1<⋯<tn<∞, we define(3)E=ψWt1,Wt2−Wt1,…,Wtn−Wtn−1≔ψn,where ψn is defined iteratively by(4)ψ1y1,y2,…,yn−1=Eψy1,y2,…,ym−1,Wtn−Wtn−1,ψ2y1,y2,…,yn−2=Eψ1y1,y2,…,ym−2,Wtn−1−Wtn−2,⋮ψn−1y1=Eψn−2y1,Wt2−Wt1,ψny1=Eψn−1Wt1.The conditional expectation ofϑω=ψWt1,Wt2−Wt1,…,Wtn−Wtn−1 is defined as(5)Eϑℱti≔ψn−iWt1,Wt2−Wt1,…,Wti−Wti−1.Definition 1.
(G-Brownian motion). The expectation operatorE (a nonlinear operator) defined above is called G-expectation, and the corresponding coordinate process W is called G-Brownian motion.
Forϑ∈LlipℱTand1≤r≤∞, we denote byLGrℱT (resp. LGrℱ as the completion of LlipℱT (resp. Llipℱunder the Banach normϑr=Eϑr1/r, for ϑ. According to Denis et al. [37], LG1ℱ can be written as the collection of all the quasi-continuous random vectors X∈L0Ω with limn⟶+∞E^X|X|>n=0. Moreover, for all X∈LG1ℱ, it holds that EX=E^X.
Under the above preparation, we introduceWt to be G-Brownian motion defined on the space of G-expectation Ω,Llipℱ,E^ with quadratic variation [20]:(6)W,Ws≔Ws2−2∫0sWτdWτ,s∈0,T.Definition 2.
(see [37]). Let DΩ be the Borel σ-algebra of Ω. The capacity c^. associated with ℙ, a weakly compact class of probability measures P defined on Ω,DΩ, is defined as(7)c^B=supP∈ℙPB,B∈DΩ.
We take this lemma from [21].Lemma 1.
Assumeϒ∈LG1FT, then for some positive r and each β>0, we have(8)cϒt>β≤E^ϒtrβr,where E^Υtr<∞.Definition 3.
Assumingr⩾1 and positive T, the space of simple processes MGr,00,T is defined as(9)MGr,00,T=ητω=ητ,ω=∑i=1N−1ϑτiωIτi−1,τiτ;ϑτiω∈LGrℱτi,∀N⩾1,τi∈0,T,i=0,1…,N−1,where MGr0,T denotes the completion of MGr,00,T under this norm(10)ηMGr0,T≔1T∫0TE^ηsrds1/r.
This lemma in [25] is needed.Lemma 2.
Assumer⩾1 and ϕ∈MGr0,T, then for all t1,t2∈0,T, t1⩽t2, we have(11)E^supt1⩽t⩽u⩽t2∫t1uϕsdW,Wsr⩽t2−t1r−1∫t1t2E^ϕsrds.Lemma 3 (see [34]).
Assumingr⩾1 and ϕ∈MG20,T, we conclude(12)E^sup0⩽v⩽T∫0vϕτdWτr⩽CrE^∫0Tϕτ2dW,Wτr/2.


### 2.2. Model Preparation

This work focuses on MISDEGs interpreted as(13)dZt+∂φZt∋ft,Ztdt+ht,ZtdW,Wt+σt,ZtdWt,t≠tj,ΔZtj=Ztj+−Ztj−=IjZtj−,t=tj,j∈ℕ,Z0=Z0∈Domφ¯,where φ:ℝd⟶−∞,+∞ is a convex and lower semicontinuous function with the domain Domφ=℘∈ℝd:φ℘<∞ such that 0∈IntDomφ≠∅, and φ℘⩾φ0=0, for every ℘∈ℝd. Functions f:0,∞×ℝd⟶ℝd, h:0,∞×ℝd⟶ℝd×m, and σ:0,∞×ℝd⟶ℝd×m are continuous. Ij:ℝd⟶ℝd and impulsive moments tj satisfy 0<t1<t2<⋯tj<⋯, and limj⟶+∞tj=∞. Ztj− and Ztj+ are the left and right limits for the continuous process Zt at time tj, respectively. Wt is a m-dimensional G-Brownian motion with quadratic variation W,Wt.Definition 4.
A subdifferential operator∂φς=ι∈ℝd:ι,υ−ς+φς⩽φυ,∀υ∈ℝd with Dom∂φ=ς∈ℝd:∂φς≠∅(i)
It is called monotone if(14)ι∗−υ∗,ι−υ⩾0,∀ι,ι∗,υ,υ∗∈Gr∂φ,whereGr∂φ=ς,υ∈ℝd×ℝd:ς∈Dom∂φ,υ∈∂φς is the graph of the subdifferential operator ∂φ.(ii)
It is called maximal monotone if(15)ι,ι∗∈Gr∂φ⇔ι∗−υ∗,ι−υ⩾0,∀υ,υ∗∈Gr∂φ.We propose this definition of the solution to (13).Definition 5.
A couple of measurable and continuous stochastic processesZ,K are named a solution to equation (13) if(i)
Zt∈Domφ¯ and Z0=Z0, q.s.(ii)
For allt⩾0, Kt is of total variation K.t0<∞ and K0=0(iii)
For any0<t<∞, we have(16)Zt=Z0+∫0tfs,Zsds+∫0ths,ZsdW,Ws+∫0tσs,ZsdWs+∑j=1∞IjZtj−−Kt,q.s.(iv)
dKt∈∂φZtdt, q.s.
The following is an important lemma from [34].Lemma 4.
If the two pairs of stochastic processesZt,Kt and Z´t,K´t satisfy Definition 4, then(17)∫0tZs−Z´s,dKs−dK´s⩾0.


## 2.1. Notations

In this section, we first give the notion of sublinear expectation spaceΩ,ℋ,E, where Ω is a given state set and ℋ is a linear space of real valued functions defined on Ω. The space ℋ can be considered the space of random variables.AssumeΩ be the space of all continuous ℝn-valued functions ωtt∈ℝ+, with ω0=0, equipped with the distance(1)ρω1,ω2=∑i=1∞2−imaxt∈0,iωt1−ωt2⋅∧1.Then,Ω,ρ is a metric space.For allω∈Ω, we define the canonical process Wtω=ωt, t∈0,∞. The filtration generated by the canonical process Wtt≥0 is defined as ℱt=σWs,0≤s≤t. Let(2)LlipℱT≔ψWt1,Wt2−Wt1,…,Wtn−Wtn−1:0≤t1<⋯<tn≤T,ψ∈ℂb.lipℝn×n,where n≥1 and ℂb.lipℝn×ℝn refers to the space of Lipschitz-bounded functions on ℝn×ℝn and Llipℱ=∪i=1∞Llipℱi.For anyϑω=ψWt1,Wt2−Wt1,…,Wtn−Wtn−1∈Llipℱ with 0<t1<⋯<tn<∞, we define(3)E=ψWt1,Wt2−Wt1,…,Wtn−Wtn−1≔ψn,where ψn is defined iteratively by(4)ψ1y1,y2,…,yn−1=Eψy1,y2,…,ym−1,Wtn−Wtn−1,ψ2y1,y2,…,yn−2=Eψ1y1,y2,…,ym−2,Wtn−1−Wtn−2,⋮ψn−1y1=Eψn−2y1,Wt2−Wt1,ψny1=Eψn−1Wt1.The conditional expectation ofϑω=ψWt1,Wt2−Wt1,…,Wtn−Wtn−1 is defined as(5)Eϑℱti≔ψn−iWt1,Wt2−Wt1,…,Wti−Wti−1.Definition 1.
(G-Brownian motion). The expectation operatorE (a nonlinear operator) defined above is called G-expectation, and the corresponding coordinate process W is called G-Brownian motion.
Forϑ∈LlipℱTand1≤r≤∞, we denote byLGrℱT (resp. LGrℱ as the completion of LlipℱT (resp. Llipℱunder the Banach normϑr=Eϑr1/r, for ϑ. According to Denis et al. [37], LG1ℱ can be written as the collection of all the quasi-continuous random vectors X∈L0Ω with limn⟶+∞E^X|X|>n=0. Moreover, for all X∈LG1ℱ, it holds that EX=E^X.
Under the above preparation, we introduceWt to be G-Brownian motion defined on the space of G-expectation Ω,Llipℱ,E^ with quadratic variation [20]:(6)W,Ws≔Ws2−2∫0sWτdWτ,s∈0,T.Definition 2.
(see [37]). Let DΩ be the Borel σ-algebra of Ω. The capacity c^. associated with ℙ, a weakly compact class of probability measures P defined on Ω,DΩ, is defined as(7)c^B=supP∈ℙPB,B∈DΩ.
We take this lemma from [21].Lemma 1.
Assumeϒ∈LG1FT, then for some positive r and each β>0, we have(8)cϒt>β≤E^ϒtrβr,where E^Υtr<∞.Definition 3.
Assumingr⩾1 and positive T, the space of simple processes MGr,00,T is defined as(9)MGr,00,T=ητω=ητ,ω=∑i=1N−1ϑτiωIτi−1,τiτ;ϑτiω∈LGrℱτi,∀N⩾1,τi∈0,T,i=0,1…,N−1,where MGr0,T denotes the completion of MGr,00,T under this norm(10)ηMGr0,T≔1T∫0TE^ηsrds1/r.
This lemma in [25] is needed.Lemma 2.
Assumer⩾1 and ϕ∈MGr0,T, then for all t1,t2∈0,T, t1⩽t2, we have(11)E^supt1⩽t⩽u⩽t2∫t1uϕsdW,Wsr⩽t2−t1r−1∫t1t2E^ϕsrds.Lemma 3 (see [34]).
Assumingr⩾1 and ϕ∈MG20,T, we conclude(12)E^sup0⩽v⩽T∫0vϕτdWτr⩽CrE^∫0Tϕτ2dW,Wτr/2.


## 2.2. Model Preparation

This work focuses on MISDEGs interpreted as(13)dZt+∂φZt∋ft,Ztdt+ht,ZtdW,Wt+σt,ZtdWt,t≠tj,ΔZtj=Ztj+−Ztj−=IjZtj−,t=tj,j∈ℕ,Z0=Z0∈Domφ¯,where φ:ℝd⟶−∞,+∞ is a convex and lower semicontinuous function with the domain Domφ=℘∈ℝd:φ℘<∞ such that 0∈IntDomφ≠∅, and φ℘⩾φ0=0, for every ℘∈ℝd. Functions f:0,∞×ℝd⟶ℝd, h:0,∞×ℝd⟶ℝd×m, and σ:0,∞×ℝd⟶ℝd×m are continuous. Ij:ℝd⟶ℝd and impulsive moments tj satisfy 0<t1<t2<⋯tj<⋯, and limj⟶+∞tj=∞. Ztj− and Ztj+ are the left and right limits for the continuous process Zt at time tj, respectively. Wt is a m-dimensional G-Brownian motion with quadratic variation W,Wt.Definition 4.
A subdifferential operator∂φς=ι∈ℝd:ι,υ−ς+φς⩽φυ,∀υ∈ℝd with Dom∂φ=ς∈ℝd:∂φς≠∅(i)
It is called monotone if(14)ι∗−υ∗,ι−υ⩾0,∀ι,ι∗,υ,υ∗∈Gr∂φ,whereGr∂φ=ς,υ∈ℝd×ℝd:ς∈Dom∂φ,υ∈∂φς is the graph of the subdifferential operator ∂φ.(ii)
It is called maximal monotone if(15)ι,ι∗∈Gr∂φ⇔ι∗−υ∗,ι−υ⩾0,∀υ,υ∗∈Gr∂φ.We propose this definition of the solution to (13).Definition 5.
A couple of measurable and continuous stochastic processesZ,K are named a solution to equation (13) if(i)
Zt∈Domφ¯ and Z0=Z0, q.s.(ii)
For allt⩾0, Kt is of total variation K.t0<∞ and K0=0(iii)
For any0<t<∞, we have(16)Zt=Z0+∫0tfs,Zsds+∫0ths,ZsdW,Ws+∫0tσs,ZsdWs+∑j=1∞IjZtj−−Kt,q.s.(iv)
dKt∈∂φZtdt, q.s.
The following is an important lemma from [34].Lemma 4.
If the two pairs of stochastic processesZt,Kt and Z´t,K´t satisfy Definition 4, then(17)∫0tZs−Z´s,dKs−dK´s⩾0.


## 3. Periodic Averaging Principle

Here, we focus on the periodic averaging method for (13). For ε∈0,ε0 with fixed quantity ε00<ε0<1/4, we consider these initial MISDEGs:(18)dZεt=εft,Zεtdt+εht,ZεtdW,Wt+εσt,ZεtdWt−εdKt,t≠tj,ΔZεtj=εIjZεtj−,t=tj,j∈ℕ,Zε0=Z0∈Domφ¯,where f,h,σ are bounded T-periodic in the first argument. Moreover, impulsive moments are also periodic such that there exists a l∈ℕ satisfying 0⩽t1<t2<⋯∞<tl<T, and for each j>l, we obtain tj=tj−l+T and Ij=Ij−l.Assume the following simplified MSDEGs without impulses, then we get(19)dZ¯εt=εf¯Z¯εt+I¯Z¯εtdt+εh¯Z¯εtdW,Wt+εσ¯Z¯εtdWt−εdK¯t,Z¯ε0=Z¯0∈Domφ¯,and with f¯:ℝd⟶ℝd, h¯:ℝd⟶ℝd×m, σ¯:ℝd⟶ℝd×m, and I¯:ℝd⟶ℝd are measurable functions defined as(20)f¯℘=1T∫0Tfs,℘ds,h¯℘=1T∫0Ths,℘ds,σ¯℘=1T∫0Tσs,℘ds,I¯℘=1T∑j=1lIj℘.For deriving our main findings, we bring the present conditions.Hypothesis 1.
AssumeGt,℘:ℝ+×ℝ+⟶ℝ+ where(1a) For each fixedz∈0,+∞, Gt,z is locally integrable in t, and it is continuous, increasing, and concave in z for each fixed t⩾0 and Gt,0=0.(1b) For allZ1,Z2∈ℝd and t⩾0, we get(21)ft,Z1−ft,Z22+ht,Z1−ht,Z22+σt,Z1−σt,Z22⩽Gt,Z1−Z22.(1c) If a continuous and positive functionℳτ derives(22)ℳτ⩽γ∫0τGv,ℳvdv,τ∈ℝ,ℳ0=0.for any γ>0, then ℳτ≡0∀τ∈0,∞.Hypothesis 2.
For allt∈0,T, it follows that(23)ft,02+ht,02+σt,02⩽L,where L>0 is a constant.Hypothesis 3.
For allZ1,Z2∈ℝd, we find two positive constants K1,K2 satisfying(24)IjZ12⩽K1,IjZ1−IjZ22⩽K2Z1−Z22.Hypothesis 4.
For eacht∈0,∞, there exist Q>0 satisfying ft,Z2⩽Q, ht,Z2⩽Q, σt,Z2⩽Q, f¯Z2⩽Q, h¯Z2⩽Q and σ¯Z2⩽Q.Theorem 1.
Assume that Hypotheses1–3 hold, then Equation (13) has a unique solution.Proof.
According to the assumptions and use of the same argument of Propositions 3.1 and 3.2 and Theorem1 in [34], it is easy to prove that Equation (13) has a unique solution. Here, we omit the detailed proof.
Theorem2 argues that nonautonomous MSDEGs with impulses (18) can strongly be replaced by autonomous MSDEGs without impulses (19).Theorem 2.
Assume Hypotheses1–4 hold and suppose Zε and Z¯ε are solutions for Equations (18) and (19), respectively, then, for any ε1∈0,ε0 and 0≤t≤∞, M>0 exists, satisfying Z0−Z¯0≤Mε and(25)limε⟶0E⌢sup0≤s≤tZεs−Z¯εs2=0.For allε∈0,ε1.Proof.
Applying the G-Itô formula, we have(26)Zεt−Z¯εt2=Z0−Z¯02+2ε∫0tZεs−Z¯εs,fs,Zεs−f¯Z¯εsds,+2ε∫0tZεs−Z¯εs,hs,Zεs−h¯Z¯εsdW,Ws+ε∫0tσs,Zεs−σ¯Z¯εs2dW,Ws,+2ε∫0tZεs−Z¯εs,σs,Zεs−σ¯Z¯εsdWs−2ε∫0tZεs−Z¯εs,dKs−dK¯s,+2εZεt−Z¯εt,∑j=1∞IjZεtj−−∫0tI¯Z¯εsds.
It is obvious from Lemma4 that(27)2ε∫0tZεs−Z¯εs,dKs−dK¯s⩾0.
Therefore, we have(28)sup0⩽s⩽tZεs−Z¯εs2,⩽Z0−Z¯02+2εsup0⩽s⩽t∫0sZετ−Z¯ετ,fτ,Zετ−f¯Z¯ετdτ,+2εsup0⩽s⩽t∫0sZετ−Z¯ετ,hτ,Zετ−h¯Z¯ετdW,Wτ,+εsup0⩽s⩽t∫0sστ,Zετ−σ¯Z¯ετ2dW,Wτ,+2εsup0⩽s⩽t∫0sZετ−Z¯ετ,στ,Zετ−σ¯Z¯ετdWτ,+2εsup0⩽s⩽tZεs−Z¯εs,∑j=1∞IjZεtj−−∫0sI¯Z¯ετdτ≔Z0−Z¯02+∑i=15Ji.
Now, the technique of plus and minus gives(29)E^J1=2εE^sup0⩽s⩽t∫0sZετ−Z¯ετ,fτ,Zετ−f¯Z¯ετdτ,⩽2εE^sup0⩽s⩽t∫0sZετ−Z¯ετ,fτ,Zετ−fτ,Z¯ετdτ,+2εE^sup0⩽s⩽t∫0sZετ−Z¯ετ,fτ,Z¯ετ−f¯Z¯ετdτ≔J11+J12.
Due to Young’s inequality and Hypothesis (1b), we conclude(30)J11⩽2εE^sup0⩽s⩽t∫0sZετ−Z¯ετ,fτ,Zετ−fτ,Z¯ετdτ,⩽ε2E^sup0⩽s⩽t∫0sZετ−Z¯ετ2dτ+E^sup0⩽s⩽t∫0sfτ,Zετ−fτ,Z¯ετ2dτ,⩽ε2E^∫0tZετ−Z¯ετ2dτ+E^∫0tGτ,Zετ−Z¯ετ2dτ.
Lettingq be large so that qT⩽t, we may obtain by Hypotheses (1b) and 4:(31)J12=2εE^sup0⩽s⩽t∫0sZετ−Z¯ετ,fτ,Z¯ετ−f¯Z¯ετdτ,⩽εE^∫0tZετ−Z¯ετ2dτ+εE^sup0⩽s⩽t∫0sfτ,Z¯ετ−f¯Z¯ετ2dτ,⩽εE^∫0tZετ−Z¯ετ2dτ+εE^sup0⩽s⩽t∑l=1q∫l−1TlTfτ,Z¯ετ−f¯Z¯ετ2dτ,+εE^sup0⩽s⩽t∫qTsfτ,Z¯ετ−f¯Z¯ετ2dτ,⩽3εE^sup0⩽s⩽t∑l=1q∫l−1TlTfτ,Z¯ετ−fτ,Z¯εlT2dτ,+3εE^sup0⩽s⩽t∑l=1q∫l−1TlTfτ,Z¯εlT−f¯Z¯εlT2dτ,+3εE^sup0⩽s⩽t∑l=1q∫l−1TlTf¯Z¯εlT−f¯Z¯ετ2dτ,+εE^sup0⩽s⩽t∫qTsfτ,Z¯ετ−f¯Z¯ετ2dτ+εE^∫0tZετ−Z¯ετ2dτ,⩽εE^∫0tZετ−Z¯ετ2dτ+6qεE^∫0TGτ,Z¯ετ−Z¯εiT2ds+12qεTQ+4εQT.
For eachl=1,…,q.
Consequently, we deduce(32)E^J1⩽ε1+εE^∫0tZετ−Z¯ετ2dτ+E^∫0tGτ,Zετ−Z¯ετ2dτ+6qεE^∫0TGτ,Z¯ετ−Z¯εlT2dτ+12qεTQ+4εQT.
ForJ2, we get with the aid of Lemma 2:(33)E^J2=2εE^sup0⩽s⩽t∫0sZετ−Z¯ετ,hτ,Zετ−h¯Z¯ετdW,Wτ,⩽2εCE^∫0tZετ−Z¯ετ,hτ,Zετ−h¯Z¯ετdτ,⩽2εCE^∫0tZετ−Z¯ετ,hτ,Zετ−hτ,Z¯ετdτ,+2εCE^∫0tZετ−Z¯ετ,hτ,Z¯ετ−h¯Z¯ετdτ≔J21+J22.
Similar toJ11, hawse have(34)J21=2εCE^sup0⩽s⩽t∫0sZετ−Z¯ετ,hτ,Zετ−hτ,Z¯ετdτ,⩽εCE^∫0tZετ−Z¯ετ2du+CE^∫0thτ,Zετ−hτ,Z¯ετ2dτ,⩽εCE^∫0tZετ−Z¯ετ2du+CE^∫0tGτ,Zετ−Z¯ετ2dτ.
According toJ12, J22 becomes(35)J22⩽εCE^∫0tZετ−Z¯ετ2dτ+εCE^sup0⩽s⩽t∫0shτ,Z¯ετ−h¯Z¯ετ2dτ,⩽εCE^sup0⩽s⩽t∑l=1q∫l−1TlThτ,Z¯ετ−h¯Z¯ετ2dτ,+εCE^sup0⩽s⩽t∫qTshτ,Z¯ετ−h¯Z¯ετ2dτ+εCE^∫0tZετ−Z¯ετ2dτ,⩽3εCE^sup0⩽s⩽t∑l=1q∫l−1TlThτ,Z¯ετ−hτ,Z¯εlT2dτ,+3εCE^sup0⩽s⩽t∑l=1q∫l−1TlThτ,Z¯εlT−h¯Z¯εlT2dτ,+3εCE^sup0⩽s⩽t∑l=1q∫l−1TlTh¯Z¯εlT−h¯Z¯ετ2dτ,+εCE^sup0⩽s⩽t∫qTshτ,Z¯ετ−h¯Z¯ετ2dτ+εCE^∫0tZετ−Z¯ετ2dτ,⩽6εCqE^∫0TGτ,Z¯ετ−Z¯εlT2dt+εCE^∫0tZετ−Z¯ετ2dτ+12εCqTQ+4εCQT.
Then, we have(36)E^J2⩽Cε+εE^∫0tZετ−Z¯ετ2dτ+CE^∫0tGτ,Zετ−Z¯ετ2dτ,+6εCqE^∫0TGτ,Z¯ετ−Z¯εlT2dτ+12εCqTQ+4εCQT.
Similarly, it can be deduced that(37)E^J3⩽2εCE^∫0tGτ,Zετ−Z¯ετ2dτ+24εCqTQ+8εCQT,+12εCqE^∫0TGτ,Z¯ετ−Z¯εlT2dτ.
By Lemma3 and the inequality of Young, it can be obtained that(38)E^J4=2εE^sup0⩽s⩽t∫0sZετ−Z¯ετ,στ,Zετ−σ¯Z¯ετdWτ,⩽2εCE^∫0tZετ−Z¯ετ,στ,Zετ−σ¯Z¯ετ2dW,Wτ1/2,⩽2εCE^sup0⩽u⩽tZετ−Z¯ετ∫0tστ,Zετ−σ¯Z¯ετ2dW,Wτ1/2,⩽εE^sup0⩽τ⩽tZετ−Z¯ετ2+εCE^∫0tστ,Zετ−σ¯Z¯ετ2dW,Wτ.
Similar to (37), we get(39)E^J4⩽εE^sup0⩽τ⩽tZετ−Z¯ετ2+2εCE^∫0tGτ,Zετ−Z¯ετ2dτ,+12εCqE^∫0TGτ,Z¯ετ−Z¯εlT2dτ+24εCqTQ+8εCQT.
With respect toJ5, Young’s inequality and Hypothesis 2 yield(40)E^J5=2εE^sup0⩽s⩽tZεs−Z¯εs,∑j=1∞IjZεtj−∫0sI¯Z¯ετdτ,⩽εE^sup0⩽s⩽tZεs−Z¯εs2+εεE^sup0⩽s⩽t∑j=1∞IiZεtj−∫0sI¯Z¯ετdτ2,⩽εE^sup0⩽s⩽tZεs−Z¯εs2+2εεlq+1E^sup0⩽s⩽t∑j=1lIjZεtj2,+2εεq+11T2ltE^sup0⩽s⩽t∑j=1l∫0sIjZ¯ετ2dτ,⩽εE^sup0⩽s⩽tZεs−Z¯εs2+2εεq+1l2K1+2εεq+1l2q+12K1,⩽εE^sup0⩽s⩽tZεs−Z¯εs2+2εεq+1C´.
Taking expectation from (28) and combining with (32)–(40), we obtain(41)E^sup0⩽s⩽tZεs−Z¯εs2,⩽εε1+ε+C1+εE^∫0tZετ−Z¯ετ2dτ,+1+C+2Cε+εE^∫0tGτ,Zετ−Z¯ετ2dτ,+ε+ε6qC+12CqE^∫0TGτ,Z¯ετ−Z¯εlT2dτ,+ε+ε12CqQT+24CqQT+4CQT+8CQT,+2εεq+1C´+2εE^sup0⩽s⩽tZεs−Z¯εs2.
Thus, Jensen inequality yields(42)E^sup0⩽s⩽tZεs−Z¯εs2⩽εε1+ε+C1+ε1−2ε∫0tE^sup0⩽τ⩽sZετ−Z¯ετ2ds,+1+C+2Cε+ε1−2ε∫0tGτ,E^sup0⩽τ⩽sZετ−Z¯ετ2ds,+ε+εC11−2ε∫0TGτ,E^sup0⩽τ⩽sZ¯ετ−Z¯εlT2ds,+ε+εC2+2εεq+1C´1−2ε,where C1=6qC+12Cq and C2=12CqQT+24CqQT+4CQT+8CQT.
Taking limit asε⟶0 in Equation (42), we obtain(43)limε⟶0E^sup0⩽s⩽tZεs−Z¯εs2⩽1+C∫0tGτ,limε⟶0E^sup0⩽τ⩽sZετ−Z¯ετ2ds,which with the help of Hypothesis (1c) gives(44)limε⟶0E^sup0⩽s⩽tZεs−Z¯εs2=0.
Our proof is therefore completed.Theorem 3.
Suppose Hypotheses1–4 hold, then, for any two positive constants θ1 and θ2, there is a number ε1∈0,ε0 so that(45)E^Zεt−Z¯εt2≤θ1ε,∀t∈0,θ2ε−121−ε,where ε∈0,ε1.Proof.
Noticing that for everyt∈l−1T,lT, we have(46)Z¯εt−Z¯εlT=ε∫lTtf¯Z¯ετdτ+ε∫lTth¯Z¯ετdW,Wτ,+ε∫lTtσ¯Z¯ετdWτ−ε∫lTtdK¯τ.
By G-Itô’s formula, we imply(47)E^sup0⩽s⩽tZ¯εs−Z¯εlT2,=2εE^sup0⩽s⩽t∫lTsZ¯ετ−Z¯εlT,f¯Z¯ετdτ,+2εE^sup0⩽s⩽t∫lTsZ¯ετ−Z¯εlT,h¯Z¯ετdW,Wτ,+εE^sup0⩽s⩽t∫lTsσ¯Z¯ετ2dW,Wτ,+2εE^sup0⩽s⩽t∫lTsZ¯ετ−Z¯εlT,σ¯Z¯ετdWτ,−2εE^sup0⩽s⩽t|∫lTsZ¯ετ−Z¯εlT,dK¯τ.
Thanks to Young’s inequality, Lemmas2 and 3, and Hypothesis 4, we have(48)E^sup0⩽s⩽tZ¯εs−Z¯εlT2,⩽εE^∫lTtf¯Z¯ετ2dτ+εE^∫lTtZ¯ετ−Z¯εlT2dτ,+εCE^∫lTth¯Z¯ετ2dτ+εCE^∫lTtZ¯ετ−Z¯εlT2dτ,+CεE^∫lTtσ¯Z¯ετ2dτ+εCE^∫lTtZ¯ετ−Z¯εlT2,+CεE^sup0⩽τ⩽tσ¯Z¯ετ2−2εE^sup0⩽s⩽tZ¯ετ−Z¯εlT‖K¯T0,⩽εq+1QT+εCQT+εCQT+εCQ,+ε+εC∫lTtE^sup0⩽τ⩽s|Z¯ετ−Z¯εlT|2ds,−εE^sup0⩽τ⩽tZ¯ετ−Z¯εlT2−εE^K¯|T02,⩽εq+1QT+εCQT+εCQT+εCQ1+ε,+ε+εC1+ε∫lTtE^sup0⩽τ⩽sZ¯ετ−Z¯εlT2ds,−ε1+εE^K¯T02.
Therefore, Gronwall’s inequality produces(49)E^sup0⩽s⩽tZ¯εs−Z¯εlT2,⩽εq+1QT+εCQT+εCQT+εCQε+1,×eCε+εε+1t=A.
Because of concavity ofGt,℘, we find ϱt⩾0, νt⩾0 so that(50)Gt,℘⩽ϱt+νt℘,∫0Tϱtdt<∞,∫0Tνtdt<∞.
Inserting equations (49) and (50) into (42), we obtain(51)E^sup0≤s≤tZεs−Z¯εs2,≤εCε+11+2sup0≤t≤Tνt+ε1+ε+1+Cε−1sup0≤t≤Tνt1−2ε,×∫0tE^sup0≤τ≤sZετ−Z¯ετ2ds,+ε1+εC2+2Ctsup0≤t≤Tϱt+C1tsup0≤t≤Tϱt+Asup0≤t≤Tϱt1−2ε,+ε2εC´q+1+1+Ctε−1sup0≤t≤Tϱt1−2ε,≤ε1−2εA1∫0tE^sup0≤τ≤sZετ−Z¯ετ2ds+ε1−2εA2+A3.
By employing Gronwall’s inequality, we find(52)E^sup0⩽s⩽tZεs−Z¯εs2⩽ε1−2εA2+A3eε1−2εA1t.
We chooseθ2>0 such that for all t∈0,θ2ε−1/21−ε⊆0,∞, and suppose θ1=A2+A3/1−2εeA1θ2, we may take ε1∈0,ε0 in a sense that for any ε∈0,ε1 and 0⩽t⩽θ2ε−1/21−ε,(53)E^sup0⩽t⩽θ2ε−1/21−εZεt−Z¯εt2⩽θ1ε.
Hence, it is proved.
By Theorem3, Theorem 4 gives the convergence in capacity between Zεt and Z¯εt.Theorem 4.
Suppose Hypotheses1–4 are true, then, for arbitrarily small δ>0, there is ε1∈0,ε0 so that for ε∈0,ε1(54)limε⟶0C^sup0≤t≤θ2ε−121−εZεt−Z¯εt>δ=0,where Theorem 3 defines η2.Proof.
For anyδ>0, Lemma 1 and Theorem 3 give(55)c^supt∈0,θ2ε−1/21−εZεt−Z¯εt>δ,⩽1δ2E^supt∈0,θ2ε−1/21−εZεt−Z¯εt2⩽θ1εδ2,
which tends to 0 asεgoes to 0.Remark 1.
IfIj.≡0j∈ℕ in Equation (13), then the periodic averaging method for MSDEGs is recovered. Moreover, compared to the published work on the averaging principles for MSDEs [9–12], we started to present the periodic averaging principle to MSDEs with or without impulses.


## 4. Examples

Here, a couple of examples are brought to justify the theoretical method for MISDEGs.Example 1.
We refer to these one-dimensional linear MISDEGs as follows:(56)dZεt+∂φZεt∋2εcos2tZεtdt+2εsintZεtdW,Wt,+εXεtdWt,t≠tj,ΔZεtj=εjZεtj−,t=tj,j∈ℕ,Zε0=Z0∈Domφ¯,where Wtt⩾0 represents one-dimensional G-Brownian motion W1∼N0×σ¯2,σ¯2. Let(57)ft,Zε=2cos2tZε,ht,Zε=2sintZε,σt,Zε=Zε,IjZε=jZε.
We define(58)f¯Z¯ε=1π∫0πft,Z¯εdt=Z¯ε,h¯Z¯ε=1π∫0πht,Z¯εdt=4πZ¯ε.
And(59)σ¯Z¯ε=1π∫0πσt,Z¯εdt=Z¯ε,I¯Z¯ε=1π∑j=1lIjZ¯ε=l1+l2πZ¯ε.
The associated simplified equation is(60)dZ¯εt+∂φ¯Z¯εt∋ε1+ll+12πZ¯εtdt+4πεZ¯εtdW,Wt,+εZ¯εtdWt,Z¯ε0=Z¯0∈Domφ¯.
It seems that Equation (60) is linear MSDEGs with solution(61)Z¯εt=eε1+ll+12πt+4πε−ε22W,Wt+εWt,×Z¯0−ε∫0te−ε1+ll+12πs−4πε−ε22W,Ws−εWsdK¯s.
It is not difficult to check that Hypotheses1–4 are satisfied. Therefore, Theorems 2–4 are applicable, and the solutions of (56) and (60) are convergent in L2-sense and capacity.Example 2.
We refer to these one-dimensional nonlinear MISDEGs as follows:(62)dZεt+∂φZεt∋ερtsin2tκZεtdt+ερtcos2tκZεtdW,Wt,+2ερtsintcostκZεtdWt,t≠tj,ΔZεtj=εj2κZεtj−,t=tj,j∈ℕ,Zε0=Z0∈Domφ¯,where Wtt⩾0 represents one-dimensional G-Brownian motion W1∼N0×σ¯2,σ¯2, ρ2t>0 is a locally integrable function, and κ. is a continuous function satisfying(63)κu−κv2⩽Γu−v2,u,v∈ℝ,where Γ. is a concave function defined as(64)Γz=0,z=0,czlog℘−1,0⩽z⩽β,cβlogβ−1,z>β.
Or(65)Γz=0,z=0,czlogz−113loglogz−1,0⩽z⩽β,cβlogβ−113loglogβ−1,z>β.
Withc>0, 0<β<1 is sufficiently small, Γ0=0, and ∫0+∞1/Γzdz=+∞. We define(66)f¯Z¯ε=1π∫0πρtsin2tκZ¯εdt=ρt2κZ¯ε,h¯Z¯ε=1π∫0πρtcos2tκZ¯εdt=ρt2κZ¯ε,σ¯Z¯ε=2π∫0πρtsintcostκZ¯εtdt=−2ρtπκZ¯ε.
We have(67)I¯Z¯ε=π−1∑j=1lj2κZ¯ε,=ll+12l+16πκZ¯ε.
Therefore, the associated simplified equation is(68)dZ¯εt+∂φ¯Z¯εt∋ε2ρt+l1+l2l+13πκZ¯εtdt+ε2ρtκZ¯εtdW,Wt,−2επρtκZ¯εtdWt,Z¯ε0=Z¯0∈Domφ¯.
It is obvious that Equation (68) is not linear MSDEGs and that their analytical solution cannot be obtained. Note that(69)ft,z1−ft,z22+ht,z1−ht,z22+σt,z1−σt,z22,=sint4+cost4+2sint2cost2ρ2tκz1−κz22,⩽ρ2tΓz1−z22.
Therefore, Hypothesis1 is satisfied, and it is easy to check that Hypotheses 2–4 are also satisfied. Hence, Theorems 2–4 hold, and the solutions of Equations (62) and (68) are convergent in the mean square sense and capacity.


## 5. Conclusion

In this paper, by focusing on MSDEs with impulses and G-Brownian motion, we establish a periodic averaging principle under non-Lipschitz conditions. We proved that the solutions to simplified MSDEGs without impulses converge to those of the initial MISDEGs in mean square sense and also in capacity. For future research, a two-time scale SDE driven by G-Brownian motion under non-Lipschitz conditions is interesting and important. It deserves to be further investigated in the future.



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