Discrete‐time mean‐field stochastic linear‐quadratic optimal control problem with finite horizon

Abstract

AbstractIn this paper we consider the discrete‐time mean‐field stochastic linear‐quadratic (MF‐LQ) optimal control problem with indefinite weighting matrices. First, we establish the maximum principle, and by the solvability of mean‐field forward‐backward stochastic difference equations derived from the maximum principle, we characterize the existence of the open‐loop optimal control for the MF‐LQ problem. Then, by virtue of introducing the linear matrix inequalities condition, we obtain the solvability of the generalized difference Riccati equations (GDREs). Moreover, we show that the indefinite MF‐LQ problem is well‐posed if and only if the GDREs are solvable. Finally, a numerical example is used to show the effectiveness of the obtained results.