  In this paper, we consider a problem of contract theory in which several
Principals hire a common Agent and we study the model in the continuous time
setting. We show that optimal contracts should satisfy some equilibrium
conditions and we reduce the optimisation problem of the Principals to a system
of coupled Hamilton-Jacobi-Bellman (HJB) equations. We provide conditions
ensuring that for risk-neutral Principals, the system of coupled HJB equations
admits a solution. Further, we apply our study in a more specific
linear-quadratic model where two interacting Principals hire one common Agent.
In this continuous time model, we extend the result of Bernheim and Whinston
(1986) in which the authors compare the optimal effort of the Agent in a
non-cooperative Principals model and that in the aggregate model, by showing
that these two optimisations coincide only in the first best case. We also
study the sensibility of the optimal effort and the optimal remunerations with
respect to appetence parameters and the correlation between the projects.
