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

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Introduction
Te 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 diferential equations (SDEs) [1][2][3][4][5][6][7][8]. As one of the most signifcant models, recently, multivalued stochastic diferential 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 fuctuating 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 simplifed 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][14][15][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 simplifed autonomous SDE without impulses to replace original nonautonomous ISDEs [17][18][19].
Te theory of G-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][23][24][25][26][27][28][29][30][31][32][33]. In 2017, Ren et al. [34] studied a new model of multivalued SDEGs and satisfed 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 and G-stochastic calculus, we, in this work, present the periodic averaging principle for MSDEGs with impulses (MISDEGs) under the Taniguchi non-Lipschitz condition [36]. Tis article's contributions are highlighted as follows: (i) Te model uncertainty described by G-Brownian motion fuctuation and jumps presented by impulses show our MISDEG model's generality. (ii) It is observed that the proofs of Teorems 2.1, 3.1, and 3.9 in [10][11][12] [35], respectively, depend on the second moment boundedness property of the solution. However, in our case, Teorem 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 subdiferential term, which is diferent from the multivalued term in [9][10][11][12]. (iii) Our non-Lipschitz condition is more general than the one used in [9][10][11][12]35] and considers them as special cases. Terefore, the results in [9][10][11][12]35] are generalized and extended.
Section 2 is concerned with some preliminary notions, defnitions, lemmas, and interpretation of the MISDEG model. In Section 3, we give the approximation in capacity and L 2 -sense between the initial MISDEG and simplifed 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.

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

Notations.
In this section, we frst give the notion of sublinear expectation space (Ω, H, E), where Ω is a given state set and H is a linear space of real valued functions defned on Ω. Te space H can be considered the space of random variables.
Assume Ω be the space of all continuous R n -valued functions ω t t∈R + , with ω 0 � 0, equipped with the distance Ten, (Ω, ρ) is a metric space. For all ω ∈ Ω, we defne the canonical process W t (ω) � ω t , t ∈ [0, ∞). Te fltration generated by the canonical process where n ≥ 1 and C b.lip (R n × R n ) refers to the space of Lipschitz-bounded functions on R n × R n and where ψ n is defned iteratively by Defnition 1 (G-Brownian motion). Te expectation operator E (a nonlinear operator) defned above is called G -expectation, and the corresponding coordinate process W is called G-Brownian motion. For ϑ ∈ L lip (F T ) and 1 ≤ r ≤ ∞, we denote by L r G (F T ) (resp. L r G (F) as the completion of L lip (F T ) (resp. L lip (F) under the Banach norm ‖ϑ‖ r � (E[|ϑ| r ]) 1/r , for ϑ. According to Denis et al. [37], L 1 G (F) can be written as the collection of all the quasi-continuous random vectors . Under the above preparation, we introduce W t to be G-Brownian motion defned on the space of G-expectation (Ω, L lip (F), E) with quadratic variation [20]: Defnition 2 (see [37]). Let D(Ω) be the Borel σ-algebra of Ω . Te capacity c(.) associated with P, a weakly compact class of probability measures P defned on (Ω, D(Ω)), is defned as 2 Complexity We take this lemma from [21].
, then for some positive r and each β > 0, we have where E|Υ(t)| r < ∞.
Defnition 3. Assuming r⩾1 and positive T, the space of simple processes M r,0 where Tis lemma in [25] is needed.

Model Preparation. Tis work focuses on MISDEGs interpreted as
where φ: R d ⟶ (−∞, +∞] is a convex and lower semi- are the left and right limits for the continuous process Z(t) at time t j , respectively. W(t) is a m-dimensional G-Brownian motion with quadratic variation 〈W, W〉 t . where We propose this defnition of the solution to (13).

Complexity 3
Te following is an important lemma from [34].

Periodic Averaging Principle
Here, we focus on the periodic averaging method for (13). For ε ∈ [0, ε 0 ] with fxed quantity ε 0 (0 < ε 0 < (1/4)), we consider these initial MISDEGs: where f, h, σ are bounded T-periodic in the frst argument. Moreover, impulsive moments are also periodic such that there exists a l ∈ N satisfying 0 ⩽ t 1 < t 2 < · · · ∞ < t l < T, and for each j > l, we obtain t j � t j−l + T and I j � I j−l .

Hypothesis 2. For all t ∈ [0, T], it follows that
where L > 0 is a constant.
Hypothesis 3. For all Z 1 , Z 2 ∈ R d , we fnd two positive constants K 1 , K 2 satisfying

Theorem 1. Assume that Hypotheses 1-3 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 Teorem 1 in [34], it is easy to prove that Equation (13) has a unique solution.
Here, we omit the detailed proof.
Proof. Applying the G-Itô formula, we have

(26)
It is obvious from Lemma 4 that Terefore, we have Complexity 5 Now, the technique of plus and minus gives Due to Young's inequality and Hypothesis (1b), we conclude Letting q be large so that qT⩽t, we may obtain by Hypotheses (1b) and 4:
Consequently, we deduce For J 2 , we get with the aid of Lemma 2: Similar to J 11 , hawse have According to J 12 , J 22 becomes Complexity Ten, we have Similarly, it can be deduced that By Lemma 3 and the inequality of Young, it can be obtained that Similar to (37), we get 8 Complexity With respect to J 5 , Young's inequality and Hypothesis 2 yield Taking expectation from (28) and combining with (32)-(40), we obtain (41)