A Linear-Quadratic Stackelberg Differential Game with Mixed Deterministic and Stochastic Controls

This paper is concerned with a linear-quadratic (LQ) leader-follower differential game with mixed deterministic and stochastic controls. In the game, the follower is a random controller which means that the follower can choose adapted stochastic processes, while the leader is a deterministic controller which means that the leader can choose only deterministic time functions. Such problem is motivated by a pension fund insurance problem, with government, supervisory or employer being a deterministic leader and individual producer or retail investor being a random follower. An open-loop Stackelberg equilibrium solution is considered. First, an optimal control process of the follower is characterized by a stationary condition of forward-backward stochastic differential equation (FBSDE) and a convexity condition of SDE. Then it is represented as a linear functional of optimal state variable of the follower and the leader's control variable, via a classical Riccati equation. Then an optimal control function of the leader is first characterized by a convexity condition of FBSDE and a stationary condition of mean-field type FBSDE. And it is represented as a functional of expectation of optimal state variable of the leader, with the help of a system consisting of two cross-coupled Riccati equations and a two-point boundary value problem of ordinary differential equations (ODEs). The solvabilities of this new system of Riccati equations and two-point boundary value problem and investigated.