A mathematical treatment of bank monitoring incentives

In this paper, we take up the analysis of a principal/agent model with moral hazard introduced by Pages (J. Financ. Intermed.), 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, 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. Copyright Springer-Verlag Berlin Heidelberg 2014