Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth

Tao Hao, Ying Hu, Shanjian Tang, Jiaqiang Wen

Published 2022-11-10Version 1

In this paper, we study the general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, the existence and uniqueness of local and global solutions are proved with some new ideas for one-dimensional mean-field BSDEs when the generator $g\big(t,Y,Z,\mathbb{P}_{(Y,Z)}\big)$ grows in $Z$ quadratically and the terminal value is bounded. Second, a comparison theorem for the general mean-field BSDEs is obtained with the Girsanov transform. Third, we prove the convergence of the particle systems to the mean-field BSDEs with quadratic growth and give the rate of convergence. Finally, in this framework, we use the mean-field BSDE to give a probabilistic representation for the viscosity solution of a nonlocal partial differential equation (PDE, for short), as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE.