  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, which covers the seminal Sannikov's model, where the agent can
also quit by optimally choosing the termination time of the contract. 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 Sannikov's
approach, further developed by Cvitanic, Possamai, and Touzi. 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 SDE.
