Backward SDE characterization of the finite horizon Principal-Agent problem

Abstract: We consider the finite horizon continuous-time Principal--Agent problem under deterministic discount factors. Following the Sannikov reduction to a stochastic control problem, we provide a further characterization of the Principal's value function in terms of a backward SDE inducing the corresponding optimal contract. In particular, this allows to bypass the fully nonlinear HJB equation satisfied by the Principal value function in the Markovian setting. This new approach allows to handle a new class of Principal-Agent problems which was not accessible with the existing method, namely the setting where the Agent faces a regime-switching control problem.