A continuous-time dynamic contracting problem with limited liability and finite horizon

2026-09-16

AI Summary

研究了具有有限责任和有限时间的连续时间动态契约问题，通过概率方法证明了委托人价值函数是特定PDE的唯一有界经典解。

Abstract

We perform a detailed study of a principal--agent problem in a continuous time version of the celebrated Holmström--Milgrom model (Econometrica 55 (2), 1987) where we add limited liability for the Agent. We develop a probabilistic methodology to prove that the Principal's value function is the unique bounded classical solution to a fully nonlinear and fully degenerate partial differential equation (PDE) with Cauchy-Dirichlet boundary conditions on $[0,T]\times[0,\infty)$. Indeed, we also prove infinite continuous differentiability of the solution in the interior of the domain. The strength of our regularity result is such that we can ensure existence of optimal controls in strong form---a rare occurrence in dynamic contracting---and we obtain fine properties of the optimal control map, including a characterisation via a further nonlinear degenerate PDE.

Problem

• continuous-time
• dynamic contracting
• limited liability
• finite horizon
• principal-agent problem

Innovation

• continuous-time dynamic contracting
• limited liability
• fully nonlinear PDE
• strong form optimal controls