A risk based approach to the principal-agent problem

Purpose: We aim to generalize the continuous-time principal-agent problem to incorporate time-inconsistent utility functions, such as those of mean-variance type, which are prevalent in risk management and finance. Design/methodology/approach: We use recent advancements of the Pontryagin maximum principle for forward-backward stochastic differential equations (FBSDEs) to develop a method for characterizing optimal contracts in such models. This approach addresses the challenges posed by the non-applicability of the classical Hamilton-Jacobi-Bellman equation due to time inconsistency. Findings: We provide a framework for deriving optimal contracts in the principal-agent problem under hidden action, specifically tailored for time-inconsistent utilities. This is illustrated through a fully solved example in the linear-quadratic setting, demonstrating the practical applicability of the method. Originality/value: The work contributes to the existing literature by presenting a novel mathematical approach to a class of continuous time principal-agent problems, particularly under hidden action with time-inconsistent utilities, a scenario not previously addressed. The results offer potential insights for both theoretical development and practical applications in finance and economics.