  This paper studies optimal Public Private Partnerships contract between a
public entity and a consortium, in continuous-time and with a continuous
payment, with the possibility for the public to stop the contract. The public
("she") pays a continuous rent to the consortium ("he"), while the latter gives
a best response characterized by his effort. This effect impacts the drift of
the social welfare, until a terminal date decided by the public when she stops
the contract and gives compensation to the consortium. Usually, the public can
not observe the effort done by the consortium, leading to a principal agent's
problem with moral hazard. We solve this optimal stochastic control with
optimal stopping problem in this context of moral hazard. The public value
function is characterized by the solution of an associated Hamilton Jacobi
Bellman Variational Inequality. The public value function and the optimal
effort and rent processes are computed numerically by using the Howard
algorithm. In particular, the impact of the social welfare's volatility on the
optimal contract is studied.
