Forcing and duality-corrected contracts for volatility control

Abstract

In this paper, we revisit the construction of optimal incentives in continuous-time principal–agent problems with drift and volatility control. Originally, a general approach relying on dynamic programming and second-order backward stochastic differential equations (2BSDEs) was developed by Cvitanić, Possamaï, and Touzi (2018) [8] to determine the optimal form of contracts in this setting. More recently, Chiusolo and Hubert (2026) [5] proposed a BSDE-based approach by introducing an alternative ‘contractible-volatility’ problem for the principal. In addition to the proposed new method, this work highlights that the optimality result of [8] actually hinges on an assumption—stated below as Assumption 2.3—which may not hold in general. Motivated by this, we introduce in this paper a more general class of contracts, parametrised by a function ψ subject to conditions that make the contract revealing for the agent and without loss of generality for the principal. We further provide two natural specifications of ψ: one, inspired by the BSDE approach, yielding a forcing-type contract; the other, motivated by the 2BSDE approach, correcting the duality gap when Assumption 2.3 is not satisfied.