Abstract

This talk is devoted to a Stackelberg stochastic differential game for a linear mean-field type stochastic differential system with a mean-field type quadratic cost functional in finite horizon. The coefficients in the state equation and the weighting matrices in the cost functional are all deterministic. Closed-loop Stackelberg equilibrium strategies are introduced which require to be independent of initial states. The follower's problem is solved firstly, which is a stochastic linear-quadratic optimal control problem. By converting the original problem into a new one whose optimal control is known, the closed-loop optimal strategy of the follower is characterized by two coupled Riccati equations as well as a linear mean-field type BSDE. Then the leader turns to solve a stochastic linear-quadratic optimal control problem for a mean-field type FBSDE. Necessary conditions for the existence of closed-loop optimal strategies for the leader is given by the existence of two coupled Riccati equations with a linear mean-field type BSDE. Moreover, the leader's value function is expressed via two BSDEs and two Lyapunov equations.

Joint work with Zixuan Li.