Indefinite linear quadratic control of mean-field backward stochastic differential equation

This paper is concerned with a general linear quadratic (LQ) control problem of mean-field backward stochastic differential equation (BSDE). Here, the weighting matrices in cost functional are allowed to be indefinite. Necessary and sufficient conditions for optimality are obtained via a mean-field backward-forward stochastic differential equation (BFSDE). By investigating the connections with LQ problems of mean-field forward systems and taking some limiting procedures, we establish the solvabilities of corresponding Riccati equations when the cost functional is uniformly convex. Subsequently, we develop general procedures of constructing explicit representations of optimal control and optimal cost via two Riccati equations, a mean-field BSDE and an adjoint equation. Moreover, sufficient conditions for the uniform convexity of cost functional are also proposed in terms of Riccati equations, which have not been considered in existing literatures for backward systems. Two examples are provided to illustrate our theoretical results.