Existence and Uniqueness of the Solutions to High-Dimensional McKean-Vlasov SDEs Under Non-Lipschitz Conditions

MA Li; SUN Fangfang

doi:10.21656/1000-0887.440010

Volume 44 Issue 10

Oct. 2023

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MA Li, SUN Fangfang. Existence and Uniqueness of the Solutions to High-Dimensional McKean-Vlasov SDEs Under Non-Lipschitz Conditions[J]. Applied Mathematics and Mechanics, 2023, 44(10): 1272-1290. doi: 10.21656/1000-0887.440010

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Existence and Uniqueness of the Solutions to High-Dimensional McKean-Vlasov SDEs Under Non-Lipschitz Conditions

doi: 10.21656/1000-0887.440010

MA Li^{1, 2
,},
SUN Fangfang^{1
,
,}

1.
School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, P.R.China
2.
Key Laboratory of Data Science and Smart Education, Ministry of Education, Hainan Normal University, Haikou 571158, P.R.China

Received Date: 2023-01-10
Rev Recd Date: 2023-03-11
Publish Date: 2023-10-31

Abstract

Abstract

The existence and uniqueness of the solutions to high-dimensional McKean-Vlasov stochastic differential equations with discontinuous drift coefficients and corresponding particle systems, were investigated. With the drift coefficient being piecewise Lipschitz continuous about the space variable, through Zvonkin's transformation, the original equation was converted into a new McKean-Vlasov stochastic differential equation with Lipschitz continuous coefficients. Therefore, the new equation has a unique solution. Moreover, the existence and Lipschitz continuity of the inverse function were proven according to the transformation function characteristics. Finally, based on the It's formula and the inverse function characteristics, the existence and uniqueness of the solutions to the McKean-Vlasov stochastic differential equation and the corresponding particle system were obtained.

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References

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