Linear Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations

TL;DR

This paper develops a comprehensive solution framework for linear quadratic optimal control problems involving mean-field backward stochastic differential equations with deterministic coefficients, deriving explicit formulas for optimal controls.

Contribution

It introduces a novel approach to solve mean-field backward stochastic control problems by decoupling the optimality system into Riccati equations and an MF-BSDE, ensuring unique solvability.

Findings

Coupled Riccati equations are uniquely solvable.

Explicit representation of the optimal control is obtained.

The method applies to mean-field backward stochastic differential equations with deterministic coefficients.

Abstract

This paper is concerned with linear quadratic optimal control problems for mean-field backward stochastic differential equations (MF-BSDEs, for short) with deterministic coefficients. The optimality system, which is a linear mean-field forward-backward stochastic differential equation with constraint, is obtained by a variational method. By decoupling the optimality system, two coupled Riccati equations and an MF-BSDE are derived. It turns out that the coupled two Riccati equations are uniquely solvable. Then a complete and explicit representation is obtained for the optimal control.