Random Horizon Principal-Agent Problems

We consider a general formulation of the random horizon principal-agent problem with a continuous payment and a lump-sum payment at termination. In the European version of the problem, the random horizon is chosen solely by the principal with no other possible action from the agent than exerting effort on the dynamics of the output process. We also consider the American version of the contract, where the agent can also quit by optimally choosing the termination time of the contract. Our main result reduces such nonzero-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 Sannikov [Rev. Econom. Stud., 75 (2008), pp. 957-984] approach, further developed in [J. Cvitanic, D. Possamai}, and N. Touzi, Finance Stoch., 22 (2018), pp. 1-37]. We first introduce an appropriate class of contracts for which the agent's optimal effort is immediately characterized by the standard verification argument in stochastic control theory. We then show that this class of contracts is dense in an appropriate sense, so that the optimization over this restricted family of contracts represents no loss of generality. The result is obtained by using the recent well-posedness result of random horizon second-order backward SDEs in [Y. Lin, Z. Ren, N. Touzi, and J. Yang, Electron. J. Probab., 25 (2020), 99].