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

Optimal stopping contract for Public Private Partnerships under moral hazard