Principal-agent problems with exit options

We consider the problem of when to deliver the contract payoff, in a continuous-time Principal-Agent setting, in the presence of moral hazard and/or adverse selection. The principal can design contracts of a simple form that induce the agent to ask for the payoff at the time of principal's choosing. The optimal time of payment depends on the agent's and the principal's outside options. Examples when the optimal time is random include the case when the agent can be fired, after having been paid a severance payment, and then replaced by another agent; and the case when the agent and the principal have asymmetric beliefs on the return of the output. In the case of adverse selection, the agents of lower type are paid early, while the agents of higher type wait until the end. The methodology we use is the stochastic maximum principle and its link to Forward-Backward Stochastic Differential Equations.