# A mathematical treatment of bank monitoring incentives

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## Abstract

In this paper, we take up the analysis of a principal/agent model with moral hazard introduced by Pagès (J. Financ. Intermed. doi: 10.1016/j.jfi.2012.06.001, 2012), with optimal contracting between competitive investors and an impatient bank monitoring a pool of long-term loans subject to Markovian contagion. We provide here a comprehensive mathematical formulation of the model and show, using martingale arguments in the spirit of Sannikov (Rev. Econ. Stud. 75:957–984, 2008), how the maximization problem with implicit constraints faced by investors can be reduced to a classical stochastic control problem. The approach has the advantage of avoiding the more general techniques based on forward-backward stochastic differential equations described by Cvitanić and Zhang (Contract Theory in Continuous Time Models, Springer 2012) and leads to a simple recursive system of Hamilton–Jacobi–Bellman equations. We provide a solution to our problem by a verification argument and give an explicit description of both the value function and the optimal contract. Finally, we study the limit case where the bank is no longer impatient.

## Keywords

Principal/agent problem Dynamic moral hazard Optimal incentives Optimal securitization Stochastic control Verification theorem## Mathematics Subject Classification (2000)

60H30 91G40## JEL Classification

G21 G28 G32## Notes

### Acknowledgements

Research partly supported by the Chair *Financial Risks* of the *Risk Foundation* sponsored by Société Générale, the Chair *Derivatives of the Future* sponsored by the Fédération Bancaire Française, and the Chair *Finance and Sustainable Development* sponsored by EDF and Calyon.

The authors would like to thank Nizar Touzi for his precious advice, as well as two anonymous referees and an associate editor who helped to improve a previous version of this paper.

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