Continuous-time incentives in hierarchies

This paper studies continuous-time optimal contracting in a hierarchy, generalising the model of Sung (Math. Financ. Econ. 9:195-213, 2015). More precisely, in this hierarchical model, the principal (she) can contract with a manager (he) to incentivise him to act in her best interest, despite only observing the net benefits of the total hierarchy. The manager in turn subcontracts with the agents below him. Both the agents and the manager independently control in continuous time a stochastic process rep-resenting their outcome. First, we show through this continuous-time adaptation of Sung's model that even if the agents only control the drift of their outcome, their manager controls the volatility of their continuation utility by choosing their con-tract sensitivities. This first illustrative example justifies the use of recent results by Cvitanic et al. (Finance Stoch. 22:1-37, 2018) on optimal contracting for drift and volatility control to carefully study continuous-time incentive problems in hierarchy. Some technical and numerical comparisons are provided to highlight the differences with Sung's model. Then, in a second more theoretical part, we provide the method-ology to tackle a more general hierarchy model. The solution is based on the theory of second-order backward stochastic differential equations (2BSDEs), and extends the results in (Cvitanic et al. in Finance Stoch. 22:1-37, 2018) to a multitude of agents with non-trivial interactions, especially concerning volatility control.