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 agents 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 welfares volatility on the optimal contract is studied.
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