Multi-dimensional reflected McKean-Vlasov BSDEs with the obstacle depending on both the first unknown and its distribution

TL;DR

This paper introduces and analyzes multi-dimensional reflected McKean-Vlasov backward stochastic differential equations with obstacles depending on both the process and its distribution, establishing foundational results and connections to PDEs.

Contribution

It establishes existence, uniqueness, stability, and propagation of chaos for these complex MF-RBSDEs, linking them to obstacle PDEs in Wasserstein space.

Findings

Proved existence and uniqueness of solutions.

Established propagation of chaos for particle systems.

Connected MF-RBSDEs to obstacle PDEs in Wasserstein space.

Abstract

The paper studies a multi-dimensional mean-field reflected backward stochastic differential equation (MF-RBSDE) with a reflection constraint depending on both the value process $Y$ and its distribution $[Y]$. We establish the existence, uniqueness and the stability of the solution of MF-RBSDE. We also investigate the associated interacting particle systems of RBSDEs and prove a propagation of chaos result. Lastly, we investigate the relationship between MF-RBSDE and an obstacle problem for partial differential equations in Wasserstein space within a Markovian framework. Our work provides a connection between the work of Briand et al. (2020) on BSDEs with normal reflection in law and the work of Gegout-Petit and Pardoux (1996) on classical multi-dimensional RBSDEs.