Mean-field backward stochastic differential equations with conditional reflection

Yanrong Chang; Heng Du; Yanrong Chang; Heng Du

doi:10.3934/math.2025950

AIMS Mathematics

2025, Volume 10, Issue 9: 21273-21286. doi: 10.3934/math.2025950

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Research article

Mean-field backward stochastic differential equations with conditional reflection

Yanrong Chang ^,,
Heng Du

School of General Education, Shanxi College of Technology, Shuozhou 036000, China

Received: 22 July 2025 Revised: 08 September 2025 Accepted: 09 September 2025 Published: 16 September 2025
MSC : 60H25, 60H30

This paper investigated the well-posedness of solutions for a class of mean-field backward stochastic differential equations(BSDEs) with conditional reflection. First, based on the Skorohod lemma, we derived explicit estimates for the solutions and rigorously proved their uniqueness. For the special case where the generator was independent of the solution$ (Y, Z) $ and their distribution, we introduced the Snell envelope approach, which not only established the existence of solutions but also revealed their intrinsic connection to optimal stopping problems under partial information. Furthermore, for general generators (dependent on the solution and distribution), we proved the existence of solutions via the contraction mapping arguments. Our work extended the theoretical framework of mean-field BSDEs and provided novel analytical tools for related stochastic control problems.
- mean-field BSDEs with conditional reflection,
- the Snell envelope,
- optimal stopping,
- contraction arguments
Citation: Yanrong Chang, Heng Du. Mean-field backward stochastic differential equations with conditional reflection[J]. AIMS Mathematics, 2025, 10(9): 21273-21286. doi: 10.3934/math.2025950

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Abstract

This paper investigated the well-posedness of solutions for a class of mean-field backward stochastic differential equations(BSDEs) with conditional reflection. First, based on the Skorohod lemma, we derived explicit estimates for the solutions and rigorously proved their uniqueness. For the special case where the generator was independent of the solution$ (Y, Z) $ and their distribution, we introduced the Snell envelope approach, which not only established the existence of solutions but also revealed their intrinsic connection to optimal stopping problems under partial information. Furthermore, for general generators (dependent on the solution and distribution), we proved the existence of solutions via the contraction mapping arguments. Our work extended the theoretical framework of mean-field BSDEs and provided novel analytical tools for related stochastic control problems.

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