Mean Field Backward Stochastic Differential Equations with Double Mean Reflections
Hanwu Li, Jin Shi
TL;DR
This work develops the theory of mean-field backward SDEs with double mean reflections, where both the generator and the reflection constraints depend on the solution's law. It establishes well-posedness for Lipschitz generators via a contraction mapping based on backward Skorokhod problems and offers a penalization method for linear constraints; for quadratic generators, it proves existence and uniqueness under bounded-terminal data using BMO techniques and extends to unbounded terminals through a θ-method under convexity/concavity in Z. The results provide a robust framework for nonlocal obstacle problems and have potential implications for nonlocal PDE representations and risk-management applications. The paper also supplies penalization and supplementary proofs to broaden the methodological toolkit for doubly mean-reflected MFBSDEs.
Abstract
In this paper, we analyze the mean field backward stochastic differential equations (MFBSDEs) with double mean reflections, whose generator and constraints both depend on the distribution of the solution. When the generator is Lipschitz continuous, based on the backward Skorokhod problem with nonlinear constraints, we investigate the solvability of the doubly mean reflected MFBSDEs by constructing a contraction mapping. Furthermore, if the constraints are linear, the solution can also be constructed by a penalization method. For the case of quadratic growth, we obtain the existence and uniqueness results by using a fixed-point argument, the BMO martingale theory and the θ-method.