Backward stochastic optimal control with mixed deterministic controller and random controller and its applications in linear-quadratic control

Abstract

This paper investigates an optimal control problem driven by backward stochastic differential equation (BSDE) with two controllers – one is called deterministic controller and the other one is called random controller. Necessary and sufficient conditions for the mixed optimal control problem are derived. A linear-quadratic (LQ) case of the mixed optimal control problem is also studied. The mixed optimal controllers are explicitly expressed by the solution of a fully coupled mean-field forward–backward stochastic differential equation (SDE). One of novel features is that a kind of mean-field BSDE naturally arises from the research on the mixed optimal control problem. Finally, a product management problem is used to illustrate the theoretical results.