Convergence and Stability of the Truncated Stochastic Theta Method for McKean-Vlasov Stochastic Differential Equations Under Local Lipschitz Conditions

Chu, Hongxia; Yuan, Haiyan; Zhu, Quanxin

doi:10.3390/math13152433

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Convergence and Stability of the Truncated Stochastic Theta Method for McKean-Vlasov Stochastic Differential Equations Under Local Lipschitz Conditions

by

Hongxia Chu

¹,

Haiyan Yuan

²

and

Quanxin Zhu

^3,*

¹

School of Electrical and Information Engineering, Heilongjiang Institute of Technology, Harbin 150050, China

²

Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

³

School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(15), 2433; https://doi.org/10.3390/math13152433

Submission received: 29 June 2025 / Revised: 24 July 2025 / Accepted: 26 July 2025 / Published: 28 July 2025

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 4th Edition)

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Abstract

This paper focuses on McKean-Vlasov stochastic differential equations under local Lipschitz conditions. We first introduce the stochastic interacting particle system and prove the propagation of chaos. Then we establish a truncated stochastic theta scheme to approximate the interacting particle system and obtain the strong convergence of the continuous-time truncated stochastic theta scheme to the non-interacting particle system. Furthermore, we study the asymptotical mean square stability of the interacting particle system and the truncated stochastic theta method. Finally, we give one numerical example to verify our theoretical results.

Keywords: McKean-Vlasov stochastic differential equation; Wasserstein distance; truncated stochastic theta method; strong convergence; asymptotical mean square stability

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MDPI and ACS Style

Chu, H.; Yuan, H.; Zhu, Q. Convergence and Stability of the Truncated Stochastic Theta Method for McKean-Vlasov Stochastic Differential Equations Under Local Lipschitz Conditions. Mathematics 2025, 13, 2433. https://doi.org/10.3390/math13152433

AMA Style

Chu H, Yuan H, Zhu Q. Convergence and Stability of the Truncated Stochastic Theta Method for McKean-Vlasov Stochastic Differential Equations Under Local Lipschitz Conditions. Mathematics. 2025; 13(15):2433. https://doi.org/10.3390/math13152433

Chicago/Turabian Style

Chu, Hongxia, Haiyan Yuan, and Quanxin Zhu. 2025. "Convergence and Stability of the Truncated Stochastic Theta Method for McKean-Vlasov Stochastic Differential Equations Under Local Lipschitz Conditions" Mathematics 13, no. 15: 2433. https://doi.org/10.3390/math13152433

APA Style

Chu, H., Yuan, H., & Zhu, Q. (2025). Convergence and Stability of the Truncated Stochastic Theta Method for McKean-Vlasov Stochastic Differential Equations Under Local Lipschitz Conditions. Mathematics, 13(15), 2433. https://doi.org/10.3390/math13152433

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Convergence and Stability of the Truncated Stochastic Theta Method for McKean-Vlasov Stochastic Differential Equations Under Local Lipschitz Conditions

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