Feasible solution to discrete‐time linear quadratic stochastic Stackelberg difference game

Abstract

This paper investigates the discrete‐time linear quadratic (LQ) stochastic Stackelberg game, which has not been thoroughly addressed in previous literature. Firstly, we derive the maximum principle for the stochastic Stackelberg difference game using the variational method, and obtain the necessary and sufficient solvability conditions. However, due to the coupling between the two players and the presence of stochastic noise, obtaining explicit optimal leader and follower's strategies becomes challenging. Therefore, we present a feasible suboptimal control strategy instead. As a result, we derive a feasible suboptimal control strategy. To achieve this, we assume a linear homogeneous relationship to decouple the group of stochastic game forward‐backward stochastic differential equations (SG‐FBSDEs), which serves as a compromise for obtaining the optimal solution. With this approach, we derive a feasible solution to the stochastic Stackelberg difference game based on the solution to symmetric Riccati equations.