In this talk, we revisit the resolution of continuous-time principal–agent problems with
drift and volatility control, originally addressed by Cvitanic, Possamaı, and Touzi (2018) [1]
through dynamic programming and second-order backward stochastic differential equations
(2BSDEs), and develop new results in this framework.
We begin by introducing an alternative problem in which the principal is allowed to
directly control the quadratic variation of the output process. On the one hand, the res-
olution of this contractible-volatility problem follows the classical methodology of Sannikov
(2008) [2], thus relying on standard (first-order) BSDEs only. On the other hand, we demon-
strate that the original contract form introduced in [1] allows the principal to achieve her
contractible-volatility value, thereby ensuring both the optimality of this contract form and
the equivalence between the original and the alternative problems. At the same time, this
alternative approach reveals that the optimality of the original contract form implicitly re-
lies on an additional duality assumption, which was not identified in [1]. This observation
motivates the construction of new families of contracts that remain optimal even when the
duality assumption fails.
Altogether, this line of work both simplifies and strengthens the existing theory of continuous-
time principal–agent problems with volatility control, and opens new directions for further
extensions and applications in economics and finance.
Talk based on joint works with Alessandro Chiusolo, Dylan Possamaï, and Nizar Touzi.

References
[1] J. Cvitanic, D. Possamaı, and N. Touzi. Dynamic programming approach to principal–agent problems. Finance
and Stochastics, 22(1):1–37, 2018.
[2] Y. Sannikov. A continuous–time version of the principal–agent problem. The Review of Economic Studies, 75
(3):957–984, 2008.