Mean-field BSDEs with non-Lipschitz coefficients and double mean reflections
Li Hanwu, Shi Jin
TL;DR
We study mean-field backward SDEs with double mean reflections and non-Lipschitz generators under Mao's condition. The problem is formulated as $Y_t=\xi+\int_t^T f(s,Y_s,\mathbf{P}_{Y_s},Z_s,\mathbf{P}_{Z_s})ds-\int_t^T Z_s dB_s+K_T-K_t$ with $\mathbf{E}[L(t,Y_t)]\le 0\le \mathbf{E}[R(t,Y_t)]$, and existence/uniqueness are obtained via a Picard iteration informed by the Skorokhod problem. Under bi-Lipschitz/monotone conditions on $L,R$ and Mao-type modulus on $f$, the authors prove a unique solution $(Y,Z,K)\in \mathcal{S}^2\times \mathcal{H}^2\times BV[0,T]$, with $K=K^R-K^L$. This extends MFBSDE theory to incorporate double mean constraints with non-Lipschitz generators and provides tools for mean-field control problems with distributional obstacles.
Abstract
The present paper is devoted to the study of mean-field backward stochastic differential equations (MFBSDEs) with double mean reflections whose generators are not Lipschitz continuous. With the help of the Skorokhod problem and some a priori estimates for MFBSDEs, we establish the existence and uniqueness results for doubly mean reflected MFBSDEs.