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

Abouagwa, Mahmoud; Khalaf, Anas D.; Gul, Nadia; Alyobi, Sultan; Mohamed, Al-Sharef

doi:https://doi.org/10.1155/2022/5619693

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Abstract Introduction Preliminaries Examples Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5619693 | https://doi.org/10.1155/2022/5619693

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

Mahmoud Abouagwa,¹Anas D. Khalaf,²Nadia Gul,³Sultan Alyobi,⁴and Al-Sharef Mohamed⁵

Academic Editor: Qingdu Li

Received21 Mar 2022

Revised10 Oct 2022

Accepted25 Nov 2022

Published14 Dec 2022

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) [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 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 [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 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 [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 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 if where is the graph of the subdifferential operator .(ii)It is called maximal monotone if We 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 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 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 1–4 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 1–4 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 implyThanks to Young’s inequality, Lemmas 2 and 3, and Hypothesis 4, we haveTherefore, Gronwall’s inequality producesBecause of concavity of , we find , so thatInserting equations (49) and (50) into (42), we obtainBy employing Gronwall’s inequality, we findWe choose such that for all , and suppose , we may take in a sense that for any and ,Hence, it is proved.
By Theorem 3, Theorem 4 gives the convergence in capacity between and .

Theorem 4. Suppose Hypotheses 1–4 are true, then, for arbitrarily small , there is so that for where Theorem 3 defines .

Proof. For any , Lemma 1 and Theorem 3 givewhich tends to 0 as goes to 0.

Remark 1. If 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:where represents one-dimensional G-Brownian motion . LetWe defineAndThe associated simplified equation isIt seems that Equation (60) is linear MSDEGs with solutionIt is not difficult to check that Hypotheses 1–4 are satisfied. Therefore, Theorems 2–4 are applicable, and the solutions of (56) and (60) are convergent in -sense and capacity.

Example 2. We refer to these one-dimensional nonlinear MISDEGs as follows:where represents one-dimensional G-Brownian motion , is a locally integrable function, and is a continuous function satisfyingwhere is a concave function defined asOrWith , is sufficiently small, , and . We defineWe haveTherefore, the associated simplified equation isIt is obvious that Equation (68) is not linear MSDEGs and that their analytical solution cannot be obtained. Note thatTherefore, Hypothesis 1 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Deanship of Scientific Research at King Abdulaziz University, Jeddah. The authors, therefore, gratefully acknowledge DSR for technical support.

References

M. Abouagwa and J. Li, “Approximation properties for solutions to Itô-Doob stochastic fractional differential equations with non-Lipschitz coefficents,” Stochastics and Dynamics, vol. 19, pp. 1–21, 2019.
View at: Google Scholar
Y. Xu, J. Duan, and W. Xu, “An averaging principle for stochastic dynamical systems with Lévy noise,” Physica D, vol. 240, pp. 1395–1401, 2011.
View at: Google Scholar
Y. Xu, B. Pei, and Y. G. Li, “Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise,” Mathematical Methods in the Applied Sciences, vol. 38, no. 11, pp. 2120–2131, 2015.
View at: Publisher Site | Google Scholar
Y. Xu, B. Pei, and J. L. Wu, “Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion,” Stochastics and Dynamics, vol. 17, no. 2, Article ID 1750013, 2017.
View at: Publisher Site | Google Scholar
W. Xu, W. Xu, and S. Zhang, “The averaging principle for stochastic differential equations with Caputo fractional derivative,” Applied Mathematics Letters, vol. 93, pp. 79–84, 2019.
View at: Publisher Site | Google Scholar
M. Abouagwa, L. S. Aljoufi, R. A. R. Bantan, A. D. Khalaf, and M. Elgarhy, “Mixed neutral Caputo fractional stochastic evolution equations with infinite delay: existence, uniqueness and averaging principle,” Fractal Frac, vol. 105, pp. 1–18, 2022.
View at: Google Scholar
A. D. Khalaf, T. Saeed, R. Abu-Shanab, W. Almutiry, and M. Abouagwa, “Estimating drift parameters in a sub-fractional Vasicek-type process,” Entropy, vol. 24, no. 5, pp. 594–619, 2022.
View at: Publisher Site | Google Scholar
Q. Zhu, “Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control,” IEEE Transactions on Automatic Control, vol. 64, pp. 3764–3771, 2019.
View at: Publisher Site | Google Scholar
L. Ngoran and N. Z. Modeste, “Averaging principle for multi-valued stochastic differential equations,” Random Operators and Stochastic Equations, vol. 9, pp. 399–407, 2001.
View at: Google Scholar
J. Xu and J. Liu, “An averaging principle for multi-valued stochastic differential equations,” Stochastic Analysis and Applications, vol. 32, no. 6, pp. 962–974, 2014.
View at: Publisher Site | Google Scholar
R. Guo and B. Pei, “Stochastic averaging principles for multi-valued stochastic differential equations driven by Poisson point processes,” Stochastic Analysis and Applications, vol. 36, no. 4, pp. 751–766, 2018.
View at: Publisher Site | Google Scholar
W. Mao, L. Hu, S. You, and X. Mao, “The averaging method for multi-valued SDEs with jumps and non-Lipschitz coefficents,” Discrete and Continuous Dynamical Systems - Series B, vol. 24, pp. 4937–4954, 2019.
View at: Google Scholar
M. Abouagwa and J. Li, “Stochastic fractional differential equations driven by Lévy noise under Carathéodory conditions,” Journal of Mathematical Physics, vol. 60, no. 2, Article ID 22701, 2019.
View at: Publisher Site | Google Scholar
G. Arthi, J. H. Park, and H. Jung, “Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion,” Communications in Nonlinear Science and Numerical Simulation, vol. 32, pp. 145–157, 2016.
View at: Publisher Site | Google Scholar
S. Deng, X.-B. Shu, and J. Mao, “Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point,” Journal of Mathematical Analysis and Applications, vol. 467, no. 1, pp. 398–420, 2018.
View at: Publisher Site | Google Scholar
Y. Ren, X. Jia, and L. Hu, “Exponential stability of solutions to impulsive stochastic differential equations driven by G Brownian motion,” Discrete and Continuous Dynamical Systems - Series B, vol. 20, pp. 2157–2169, 2015.
View at: Google Scholar
S. Ma and Y. Kang, “Periodic averaging method for impulsive stochastic differential equations with Lévy noise,” Applied Mathematics Letters, vol. 93, pp. 91–97, 2019.
View at: Publisher Site | Google Scholar
P. Wang and Y. Xu, “Periodic averaging principle for neutral stochastic delay differential equations with impulses,” Complexity, vol. 2020, Article ID 6731091, 10 pages, 2020.
View at: Publisher Site | Google Scholar
A. D. Khalaf, M. Abouagwa, and X. Wang, “Periodic averaging method for impulsive stochastic dynamical systems driven by fractional Brownian motion under non-Lipschitz condition,” Advances in Difference Equations, vol. 2019, Article ID 526, 2019.
View at: Publisher Site | Google Scholar
S. E. Peng, “G Brownian motion and related stochastic calculus of Itô type,” in Stochastic Analysis and Applications, Abel Symp, vol. 2, pp. 541–567, Springer, Berlin, Germany, 2007.
View at: Google Scholar
S. Peng, “Nonlinear expectations and stochastic calculus under uncertainty, preprint,” 2010, https://arxiv.org/abs/1002.4546.
View at: Google Scholar
X. P. Bai and Y. Q. Lin, “On the existence and uniqueness of solutions to stochastic differntial equations driven by G Brownian motion with integral-Lipschitz coefficients,” Acta Mathematica Sinica, vol. 30, no. 3, pp. 589–610, 2014.
View at: Google Scholar
F. Gao, “Pathwise properties and homeomorphic flows for stochastic differential equations driven by G Brownian motion,” Stoch. Proc. Appl., vol. 119, pp. 3356–3382, 2009.
View at: Google Scholar
L. Hu, Y. Ren, and T. Xu, “P-Moment stability of solutions to stochastic differential equations driven by G Brownian motion,” Applied Mathematics and Computation, vol. 230, pp. 231–237, 2014.
View at: Google Scholar
Y. Lin, “Stochastic differential equations driven by G Brownian motion with reflecting boundary conditions,” Electronic Journal of Probability, vol. 18, pp. 1–23, 2013.
View at: Google Scholar
Q. Lin, “Some properties of stochastic differential equations driven by G Brownian motion,” Acta Math. Appl. Sin. Eng. Ser., vol. 29, pp. 923–942, 2013.
View at: Google Scholar
G. Li and Q. Yang, “Convergence and asymptotical stability of numerical solutions for neutral stochastic delay differential equations driven by G Brownian motion,” Computational and Applied Mathematics, vol. 37, pp. 4301–4320, 2018.
View at: Google Scholar
Y. Ren and L. Hu, “A note on the stochastic differential equations driven by G Brownian motion,” Statistics & Probability Letters, vol. 81, pp. 580–585, 2011.
View at: Google Scholar
Y. Ren, Q. Bi, and R. Sakthivel, “Stochastic functional differential equations with infinite delay driven by G Brownian motion,” Mathematical Methods in the Applied Sciences, vol. 36, pp. 1746–1759, 2013.
View at: Google Scholar
Y. Ren, K. Wang, and H. Yang, “Stability analysis of stochastic pantograph multi-group models with dispersal driven by G Brownian motion,” Applied Mathematics and Computation, vol. 355, pp. 356–365, 2019.
View at: Google Scholar
D. Zhang and Z. Chen, “Exponential stability for stochastic differential equation driven by G Brownian motion,” Applied Mathematics Letters, vol. 25, pp. 1906–1910, 2012.
View at: Google Scholar
M. Zhang and G. Zong, “Almost periodic solutions for stochastic differential equations driven by G Brownian motion,” Comm. Stat.-Theor. Meth., vol. 44, pp. 2371–2384, 2015.
View at: Google Scholar
Q. Zhu and T. Huang, “Stability analysis for a class of stochastic delay nonlinear systems driven by G Brownian motion,” Systems & Control Letters, vol. 140, Article ID 104699, 2020.
View at: Google Scholar
Y. Ren, J. Wang, and L. Hu, “Multi-valued stochastic differential equations driven by G Brownian motion and related stochastic control problems,” International Journal of Control, vol. 90, pp. 1132–1154, 2017.
View at: Google Scholar
X. He, S. Han, and J. Tao, “Averaging principle for SDEs of neutral type driven by G Brownian motion,” Stochastics and Dynamics, vol. 19, pp. 1–22, 2019.
View at: Google Scholar
T. Taniguchi, “Successive approximations to solutions of stochastic differential equations,” Journal of Differential Equations, vol. 96, no. 1, pp. 152–169, 1992.
View at: Publisher Site | Google Scholar
L. Denis, M. Hu, and S. Peng, “Function spaces and capacity related to a sublinear expectation: application to G Brownian motion paths,” Potential Analysis, vol. 34, pp. 139–161, 2010.
View at: Google Scholar

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Copyright © 2022 Mahmoud Abouagwa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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