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Multi-dimensional Path-dependent Forward-backward Stochastic Variational Inequalities

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Abstract

In this article, we consider a system of stochastic variational inequalities (SVIs) in the differential form. The system has a d-dimensional forward SVI X that depends on its path and carries a subdifferential operator, and a n-dimensional backward SVI coupled with X through the path of X and has another subdifferential operator. This system extends all classical stochastic processes to SVIs with general path-dependence, and enables classical SVIs with random coefficient functions. Through delicate infinite-dimensional analyses and forward-backward stochastic analyses, we establish its well-posedness of a unique strong solution under mild regularity conditions.

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Acknowledgements

The authors would like to thank anonymous reviewers and Editors for their very constructive comments and efforts on this work, which greatly improved the quality of this paper.

Funding

The research of Jing Wu was supported by NSFC (No. 12071493 and No. 11871484). The research of Ning Ning was partially supported by the Seed Fund Grant Award at Texas A&M University.

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Correspondence to Jing Wu.

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Ning, N., Wu, J. Multi-dimensional Path-dependent Forward-backward Stochastic Variational Inequalities. Set-Valued Var. Anal 31, 2 (2023). https://doi.org/10.1007/s11228-023-00665-4

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