We consider a general formulation of the Principal-Agent problem with a continuous payment and a lump-sum payment at termination, and possibly in a random horizon setting.

Our main result reduces such non-zero sum stochastic differential games to appropriate stochastic control problems which may be solved by standard methods of stochastic control theory. This reduction is obtained by following the seminal approach by Sannikov in economics, further developed by Cvitanic, Possamai & Touzi.

The approach extends to the situation where the principal is facing a crowd of interacting agents in Nash equilibrium and provides a new point of view for the optimal planning problem in mean-field games introduced by P.-L. Lions. We provide new solutions in the path-dependent context.

Nizar Touzi is Professor of applied mathematics at Ecole Polytechnique. He was previously Chair Professor at Imperial College London. He was an invited session speaker at the International Congress of Mathematicians (Hyderabad 2010). He received the Louis Bachelier prize of the French Academy of Sciences in 2012, and the Paris Europlace prize of Best young researcher in Finance in 2007. He is Co-editor and Associate Editor in various international journals in the fields of financial mathematics, applied probability, and control theory.