A Continuous-Time Dynamic Contracting Problem with Limited Liability and Finite Horizon

Andrea Bovo, Tiziano de Angelis and Stéphane Villeneuve
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Stéphane Villeneuve: TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement

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Abstract: We perform a detailed study of a principal–agent problem in a continuous time version of the celebrated Holmstr¨om–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 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.

Keywords: fully nonlinear and fully degenerate HJB équations; finite horizon; time-changed diffusions; dynamic contracting; limited liability (search for similar items in EconPapers)
Date: 2026-09-30
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