Optimal stopping contract for Public Private Partnerships under moral hazard

Ishak Hajjej; Caroline Hillairet; Mohamed Mnif

doi:10.3934/fmf.2022007

Article Contents

2022, Volume 1, Issue 4: 539-573. Doi: 10.3934/fmf.2022007 open access

This issue Previous Article On asymptotically arbitrage-free approximations of the implied volatility Next Article Pricing autocallables under local-stochastic volatility

Optimal stopping contract for Public Private Partnerships under moral hazard

1.
CREST & ENSIIE, LaMME, France
2.
CREST, ENSAE Paris, France
3.
ENIT, LAMSIN, University of Tunis El Manar, Tunis, Tunisia

^*Corresponding author: Caroline Hillairet
^*Corresponding author: Caroline Hillairet

Received: April 2021

Revised: June 2022

Early access: November 2022

Published: December 2022

This research is supported by a grant of the French National Research Agency (ANR), "Investissements d'Avenir" (LabEx Ecodec/ANR-11-LABX-0047).

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Abstract

This paper studies optimal Public Private Partnerships contracts between a public entity and a consortium, in continuous-time and with a continuous payment, and 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 effort 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 cannot observe the effort done by the consortium, leading to a principal agent's problem with moral hazard. Therefore this paper formalizes such PPP contracts into a contract theory problem. Due to the long-term characteristic of PPP contracts, the public should incentivize the consortium to provide effort not only through the terminal payment but also through the rent paid until the end of the contract. 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, 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.

Keywords:

Moral hazard,
Public Private Partnership,
stochastic control,
optimal stopping,
Hamilton Jacobi Bellman variational inequality,
Howard algorithm.

Mathematics Subject Classification: 60G40, 91B40, 91B70, 93E40.

Citation:

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Figure 1. The value function on $ [0,\bar{x}]. $

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Figure 2. Value function for different $ \sigma $

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Figure 3. Optimal rent for different $ \sigma $

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Figure 4. Optimal effort for different $ \sigma. $

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Figure 5.

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Figure 6. The consortium value function

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Figure 7. The public value function

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Figure 8. The optimal rent

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Figure 9. The optimal effort

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Table 1. The continuation region and the threshold $ x_0 $ for different $ \sigma $

$ \sigma $	Continuation region	Threshold $ x_0 $
$ 1.55 $	$ 0.390 $	0.160
$ 1.65 $	$ 0.3655 $	0.156
$ 2 $	$ 0.3005 $	0.136

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