Mean-Field Type FBSDEs under Domination-Monotonicity Conditions and Application to LQ Problems

TL;DR

This paper establishes well-posedness results for a class of mean-field forward-backward stochastic differential equations with complex coupling, extending to controlled initial and terminal states, and applies these findings to solve related linear-quadratic control problems.

Contribution

It introduces randomized domination-monotonicity conditions for MF-FBSDEs, ensuring unique solutions and extending the analysis to more general mean-field LQ control problems.

Findings

Proved well-posedness of MF-FBSDEs under new conditions.

Extended MF-LQ problems to include controlled initial and terminal states.

Derived explicit optimal controls for the extended problems.

Abstract

This paper is concerned with a class of mean-field type coupled forward-backward stochastic differential equations (MF-FBSDEs, for short), in which the coupling appears in integral terms, terminal terms, and initial terms. Inspired by various mean-field type linear-quadratic (MF-LQ,for short) optimal control problems, we proposed a type of randomized domination-monotonicity conditions, under which and the usual Lipschitz condition, we obtain a well-posedness result on MF-FBSDEs in the sense of square integrability including the unique solvability, an estimate of the solution, and the related continuous dependence property of the solution on the coefficients.The result of MF-FBSDEs in turn extends MF-LQ problems in the literature to a general situation where the initial states or the terminal states are also controlled at the same time, and gives explicit expressions of the related unique optimal controls.