A continuous-time dynamic contracting problem with limited liability and finite horizon
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]×[0,∞). 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.
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BibTeX
@misc{bovo2026continuoustime,
title = {{A continuous-time dynamic contracting problem with limited liability and finite horizon}},
author = {Andrea Bovo and Tiziano De Angelis and Stéphane Villeneuve},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18287},
eprint = {2609.18287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18287},
}
RIS
TY - UNPB
TI - A continuous-time dynamic contracting problem with limited liability and finite horizon
AU - Andrea Bovo
AU - Tiziano De Angelis
AU - Stéphane Villeneuve
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.18287
UR - https://arxiv.org/abs/2609.18287
AB - 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]×[0,∞). 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.
ER -