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 in -sense and capacity. Finally, we bring a couple of two examples to enhance our theoretical results.

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) [18]. 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) [1316]. 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 [1719].

The theory of -expectation is at the heart of our understanding of uncertainty problems, risk measures, and optimization problems [20, 21]. Peng [20] constructed the cornerstone of -expectation theory with its related random calculus, and SDEs perturbed with G-Brownian motion (SDEGs) became the subject of much systematic investigation [2233]. 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 and -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 [1012] [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 [912].(iii)Our non-Lipschitz condition is more general than the one used in [912, 35] and considers them as special cases. Therefore, the results in [912, 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 and -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 space , where is a given state set and is a linear space of real valued functions defined on . The space can be considered the space of random variables.

Assume be the space of all continuous -valued functions , with , equipped with the distance

Then, is a metric space.

For all , we define the canonical process , . The filtration generated by the canonical process is defined as . Letwhere and refers to the space of Lipschitz-bounded functions on and .

For any with , we definewhere is defined iteratively by

The conditional expectation of is defined as

Definition 1. (G-Brownian motion). The expectation operator (a nonlinear operator) defined above is called -expectation, and the corresponding coordinate process is called G-Brownian motion.
For and, we denote by (resp. as the completion of (resp. under the Banach norm, for ϑ. According to Denis et al. [37], can be written as the collection of all the quasi-continuous random vectors with . Moreover, for all , it holds that .
Under the above preparation, we introduce to be G-Brownian motion defined on the space of -expectation with quadratic variation [20]:

Definition 2. (see [37]). Let be the Borel -algebra of . The capacity associated with , a weakly compact class of probability measures defined on , is defined asWe take this lemma from [21].

Lemma 1. Assume , then for some positive r and each , we havewhere .

Definition 3. Assuming and positive , the space of simple processes is defined aswhere denotes the completion of under this normThis lemma in [25] is needed.

Lemma 2. Assume and , then for all , , we have

Lemma 3 (see [34]). Assuming and , we conclude

2.2. Model Preparation

This work focuses on MISDEGs interpreted aswhere is a convex and lower semicontinuous function with the domain such that , and , for every . Functions , , and are continuous. and impulsive moments satisfy , and . and are the left and right limits for the continuous process at time , respectively. is a -dimensional G-Brownian motion with quadratic variation .

Definition 4. A subdifferential operator with (i)It is called monotone ifwhere is the graph of the subdifferential operator .(ii)It is called maximal monotone ifWe propose this definition of the solution to (13).

Definition 5. A couple of measurable and continuous stochastic processes are named a solution to equation (13) if(i) and , q.s.(ii)For all , is of total variation and (iii)For any , we have(iv), q.s.The following is an important lemma from [34].

Lemma 4. If the two pairs of stochastic processes and satisfy Definition 4, then

3. Periodic Averaging Principle

Here, we focus on the periodic averaging method for (13). For with fixed quantity , we consider these initial MISDEGs:where are bounded -periodic in the first argument. Moreover, impulsive moments are also periodic such that there exists a satisfying , and for each , we obtain and .

Assume the following simplified MSDEGs without impulses, then we getand with , , , and are measurable functions defined as

For deriving our main findings, we bring the present conditions.

Hypothesis 1. Assume where(1a) For each fixed , is locally integrable in , and it is continuous, increasing, and concave in for each fixed and .(1b) For all and , we get(1c) If a continuous and positive function derivesfor any , then .

Hypothesis 2. For all , it follows thatwhere is a constant.

Hypothesis 3. For all , we find two positive constants satisfying

Hypothesis 4. For each , there exist satisfying , , , , and .

Theorem 1. Assume that Hypotheses 13 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 Theorem 1 in [34], it is easy to prove that Equation (13) has a unique solution. Here, we omit the detailed proof.
Theorem 2 argues that nonautonomous MSDEGs with impulses (18) can strongly be replaced by autonomous MSDEGs without impulses (19).

Theorem 2. Assume Hypotheses 14 hold and suppose and are solutions for Equations (18) and (19), respectively, then, for any and , exists, satisfying and

For all .

Proof. Applying the G-Itô formula, we haveIt is obvious from Lemma 4 thatTherefore, we haveNow, the technique of plus and minus givesDue to Young’s inequality and Hypothesis (1b), we concludeLetting be large so that , we may obtain by Hypotheses (1b) and 4:For each .
Consequently, we deduceFor , we get with the aid of Lemma 2:Similar to , hawse haveAccording to , becomesThen, we haveSimilarly, it can be deduced thatBy Lemma 3 and the inequality of Young, it can be obtained thatSimilar to (37), we getWith respect to , Young’s inequality and Hypothesis 2 yieldTaking expectation from (28) and combining with (32)–(40), we obtainThus, Jensen inequality yieldswhere and .
Taking limit as in Equation (42), we obtainwhich with the help of Hypothesis (1c) givesOur proof is therefore completed.

Theorem 3. Suppose Hypotheses 14 hold, then, for any two positive constants and , there is a number so thatwhere .

Proof. Noticing that for every , we haveBy G-Itô’s formula, we imply