The Maximum Principle for Global Solutions of Stochastic Stackelberg Differential Games

Abstract

This paper obtains the maximum principle for both stochastic (global) open-loop and stochastic (global) closed-loop Stackelberg differential games. For the closed-loop case, we use the theory of controlled forward-backward stochastic differential equations to derive the maximum principle for the leader's optimal strategy. In the special case of the open-loop linear quadratic Stackelberg game, we consider the follower's Hamiltonian system as the leader's state equation, derive the related stochastic Riccati equation, and show the existence and uniqueness of the solution to the Riccati equation under appropriate assumptions. However, for the closed-loop linear quadratic Stackelberg game, we can write the related Riccati equation consisting of forward-backward stochastic differential equations, while leaving the existence of its solution as an open problem.