Solvability and stability of mean-field stochastic differential equations driven by time-changed Lévy noise

Mahmoud Abouagwa; Maher Ibrahim Tawdrous; Mahmoud Abouagwa; Maher Ibrahim Tawdrous

doi:10.3934/math.2026657

AIMS Mathematics

2026, Volume 11, Issue 6: 15952-15989. doi: 10.3934/math.2026657

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Solvability and stability of mean-field stochastic differential equations driven by time-changed Lévy noise

Mahmoud Abouagwa ^{1,2
,
,},
Maher Ibrahim Tawdrous ^3,4

1.
College of Business, City University Ajman, Ajman 18484, United Arab Emirates
2.
Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt
3.
College of Humanities, City University Ajman, Ajman 18484, United Arab Emirates
4.
Faculty of Education, Suez Canal University, Ismailia 41522, Egypt

Received: 29 January 2026 Revised: 20 May 2026 Accepted: 26 May 2026 Published: 05 June 2026
MSC : 60B10, 60H10, 35Q83, 93E15

In this article, we focused on a class of mean-field stochastic differential equations driven by time-changed Lévy noise. We first discussed the existence and uniqueness of solutions under the non-Lipschitz case with the Lipschitz condition as the special case by adopting the Carathéodory approximation. To prove our results, we established a new time-changed retarded integral inequality, which is easy to apply in practice and can be considered as a more general tool in some situations. Then, the classical Itô formula was extended to that for mean-field stochastic differential equations driven by time-changed Lévy noise. As an application of Itô's formula, we showed that the trivial solution is $ p $-th moment asymptotically stable, stable in probability, asymptotically stable in probability, and globally asymptotically stable in probability based on the Lyapunov function. Finally, an example was presented to validate the produced results.
- mean-field stochastic differential equations,
- stability,
- time-changed Lévy noise,
- Itô's formula
Citation: Mahmoud Abouagwa, Maher Ibrahim Tawdrous. Solvability and stability of mean-field stochastic differential equations driven by time-changed Lévy noise[J]. AIMS Mathematics, 2026, 11(6): 15952-15989. doi: 10.3934/math.2026657

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Abstract

In this article, we focused on a class of mean-field stochastic differential equations driven by time-changed Lévy noise. We first discussed the existence and uniqueness of solutions under the non-Lipschitz case with the Lipschitz condition as the special case by adopting the Carathéodory approximation. To prove our results, we established a new time-changed retarded integral inequality, which is easy to apply in practice and can be considered as a more general tool in some situations. Then, the classical Itô formula was extended to that for mean-field stochastic differential equations driven by time-changed Lévy noise. As an application of Itô's formula, we showed that the trivial solution is $ p $-th moment asymptotically stable, stable in probability, asymptotically stable in probability, and globally asymptotically stable in probability based on the Lyapunov function. Finally, an example was presented to validate the produced results.

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