Résumé : The seminal work of Cvitanic, Possamai and Touzi (2018) [1] introduced a general framework for continuous-time principal-agent problems using dynamic programming and second-order backward stochastic differential equations (2BSDEs). In this talk, we first propose an alternative formulation of the principal-agent problem that allows for a more direct resolution using standard BSDEs alone. Our approach is motivated by a key observation in [1]: when the principal observes the output process X continuously, she can compute its quadratic variation pathwise. While this information is incorporated into the contract in [1], we consider here a reformulation where the principal directly controls this process in a ‘first-best' setting. The resolution of this alternative problem follows the methodology known as Sannikov's trick [2] in continuous-time principal-agent problems. We then demonstrate that the solution to this ‘first-best' formulation coincides with the original problem's solution. More specifically, leveraging the contract form introduced in [1], we establish that the ‘first-best' outcome can be attained even when the principal lacks direct control over the quadratic variation. Crucially, our approach does not require the use of 2BSDEs to prove contract optimality, as optimality naturally follows from achieving the ‘first-best' scenario. We believe that this reformulation offers a more accessible approach to solving continuous-time principal-agent problems with volatility control, facilitating broader dissemination across various fields. In the second part of the talk, we will explore how this methodology extends to more complex settings, particularly multi-agent frameworks. Research partially supported by the NSF grant DMS-2307736.

Keywords : Principal–agent problems; volatility control; moral hazard; first-best; BSDEs