A new approach to principal-agent problems with volatility control

The recent work by Cvitani\'c, Possama\"i, and Touzi (2018) [9] presents a general approach for continuous-time principal-agent problems, through dynamic programming and second-order backward stochastic differential equations (BSDEs). In this paper, we provide an alternative formulation of the principal-agent problem, which can be solved simply by relying on the theory of BSDEs. This reformulation is strongly inspired by an important remark in [9], namely that if the principal observes the output process in continuous-time, she can compute its quadratic variation pathwise. While in [9], this information is used in the contract, our reformulation consists in assuming that the principal could directly control this process, in a `first-best' fashion. The resolution approach for this alternative problem actually follows the line of the so-called `Sannikov's trick' in the literature on continuous-time principal-agent problems, as originally introduced by Sannikov (2008) [28]. We then show that the solution to this `first-best' formulation is identical to the solution of the original problem. More precisely, using the contract form introduced in [9] as `penalisation contracts', we highlight that this `first-best' scenario can be achieved even if the principal cannot directly control the quadratic variation. Nevertheless, we do not have to rely on the theory of 2BSDEs to prove that such contracts are optimal, as their optimality is ensured by showing that the `first-best' scenario is achieved. We believe that this more straightforward approach to solve continuous-time principal-agent problems with volatility control will facilitate the dissemination of these problems across many fields, and its extension to even more intricate problems.