A unified approach to global solvability for FBSDEs with diagonal generators

TL;DR

This paper presents a unified method for establishing the global solvability of multidimensional FBSDEs with diagonal generators, extending existing results and applying to stochastic control and game problems.

Contribution

It introduces a unified approach to prove the existence of a decoupling field for FBSDEs with diagonal generators under monotonicity, including Lipschitz, quadratic, and super-quadratic cases.

Findings

Existence of a decoupling field under monotonicity condition

Extensions to L^p solutions and estimates for Lipschitz case

Application to stochastic control and differential games

Abstract

In this paper, we study the global solvability of multidimensional forward-backward stochastic differential equations (FBSDEs) with diagonally Lipschitz, quadratic or super-quadratic generators. Under a certain "monotonicity" condition, we provide a unified approach which shows that there exists a decoupling field that is uniformly Lipschitz in its spatial variable. This decoupling field is closely related to bounded solution to an associated characteristic BSDE. For Lipschitz case, we provide some extensions and investigate $L^p$-solution and $L^p$ estimates. Our results gives a positive answer to a question proposed in Yong (Banach Center Publ. 122: 255-286, 2020). Applications to stochastic optimal controls and stochastic differential games are investigated.