  This article studies the problem of evaluating the information that a
Principal lacks when establishing an incentive contract with an Agent whose
effort is not observable. The Principal ("she") pays a continuous rent to the
Agent ("he"), while the latter gives a best response characterized by his
effort, until a terminal date decided by the Principal when she stops the
contract and gives compensation to the Agent. The output process of the project
is a diffusion process driven by a Brownian motion whose drift is impacted by
the Agent's effort. The first part of the paper investigates the optimal
stochastic control problem when the Principal and the Agent share the same
information. This situation, known as the first-best case, is solved by
tackling the Lagrangian problem. In the second part, the Principal observes the
output process but she may not observe the drift and the Brownian motion
separately. This situation is known as the second-best case. We derive the best
response of the Agent, then we solve the mixed optimal stopping/stochastic
control problem of the Principal under a fixed probability and on the
filtration generated by the Brownian motion, which is larger than the one
generated by the output process (that corresponds to the information available
for the Principal). Under some regularity conditions, the Principal value
function is characterized by solving the associated Hamilton Jacobi Bellman
Variational Inequality. At the optimum, we prove that the two filtrations
coincide. Finally, we compute the value of the information for the Principal
provided by the observation of the Agent's effort. It is defined as the
difference between the principal value function in the first-best and
second-best cases.
