| Abstract: |
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. |