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Well-posedness and averaging principle for non-Gaussian McKean–Vlasov stochastic differential equations with locally Lipschitz coefficients

Part of: Qualitative theory Stochastic processes Stochastic analysis Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  09 September 2025

Ying Chao ,
Jinqiao Duan ,
Ting Gao  and
Pingyuan Wei Open the ORCID record for Pingyuan Wei [Opens in a new window]
Ying Chao*
Affiliation:
Xi’an Jiaotong University
Jinqiao Duan*
Affiliation:
Great Bay University
Ting Gao*
Affiliation:
Huazhong University of Science and Technology
Pingyuan Wei*
Affiliation:
Southeast University and Peking University
*
*Postal address: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. Email: yingchao1993@xjtu.edu.cn
**Postal address: Department of Mathematics and Guangdong Provincial Key Laboratory of Mathematical and Neural Dynamical Systems, Great Bay University, Dongguan, Guangdong 523000, China. Email: duan@gbu.edu.cn
***Postal address: School of Mathematics and Statistics and Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China. Email: tgao0716@hust.edu.cn
****Postal address: School of Mathematics, Southeast University, Nanjing 211189, China; Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China. Email: weipingyuan@pku.edu.cn
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Abstract

In this paper, we investigate a class of McKean–Vlasov stochastic differential equations (SDEs) with Lévy-type perturbations. We first establish the existence and uniqueness theorem for the solutions of the McKean–Vlasov SDEs by utilizing an Eulerlike approximation. Then, under suitable conditions, we demonstrate that the solutions of the McKean–Vlasov SDEs can be approximated by the solutions of the associated averaged McKean–Vlasov SDEs in the sense of mean square convergence. In contrast to existing work, a novel feature of this study is the use of a much weaker condition, locally Lipschitz continuity in the state variables, allowing for possibly superlinearly growing drift, while maintaining linearly growing diffusion and jump coefficients. Therefore, our results apply to a broader class of McKean–Vlasov SDEs.

Keywords

McKean–Vlasov stochastic differential equationswell-posednessaveraging principleone-sided locally Lipschitz conditionLévy-type perturbations

MSC classification

Primary: 60H10: Stochastic ordinary differential equations 35Q83: Vlasov-like equations
Secondary: 60G51: Processes with independent increments; L'evy processes 34C29: Averaging method

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Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 44
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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