Mean-Field Linear-Quadratic Optimal Controls

Abstract

This chapter is concerned with a more general class of linear-quadratic optimal control problems, the mean-field linear-quadratic optimal control problem, in which the expectations of the state process and the control are involved. Two differential Riccati equations are introduced for the problem. The strongly regular solvability of these two Riccati equations is proved to be equivalent to the uniform convexity of the cost functional. In terms of the solutions to the Riccati equations, the unique optimal control is obtained as a linear feedback of the state process and its expectation. An application of the mean-field linear-quadratic optimal control theory is presented, in which analytical optimal portfolio policies are constructed for a continuous-time mean-variance portfolio selection problem. The mean-field linear-quadratic optimal control problem over an infinite horizon is also studied.