We provide a general approach to reformulating any Stackelberg game under closed-loop strategies, i.e., decisions based solely on the historical information of the output process, as a single-level optimization problem. Our first result reformulates the leader’s unconventional stochastic control problem as a standard stochastic control problem with stochastic target constraints. Thereafter, adapting results by Soner and Touzi ’02 and Bouchard, Élie, and Imbert ‘10, we show that the equilibria and the value of the game are given by the solution of a well-specified system of HJB equations. We illustrate our approach and compare it to the solutions under different information structures in the literature in an example.

Joint work with Nicolás Hernández-Santibáñez, Emma Hubert and Dylan Possamaï.