Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control

This paper is concerned with the discrete-time indefinite mean-field linear-quadratic optimal control problem. The so-called mean-field type stochastic control problems refer to the problem of incorporating the means of the state variables into the state equations and cost functionals, such as the mean-variance portfolio selection problems. A dynamic optimization problem is called to be nonseparable in the sense of dynamic programming if it is not decomposable by a stage-wise backward recursion. The classical dynamic-programming-based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies. In this paper, we show that both the well-posedness and the solvability of the indefinite mean-field linear-quadratic problem are equivalent to the solvability of two coupled constrained generalized difference Riccati equations and a constrained linear recursive equation. We characterize the optimal control set completely, and obtain a set of necessary and sufficient conditions on the mean-variance portfolio selection problem. The results established in this paper offer a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the mean-variance-type portfolio selection problems.