Contract theory in a VUCA world

In this paper we investigate a Principal-Agent problem with moral hazard under Knightian uncertainty. We extend the seminal framework of Holmstr\"om and Milgrom by combining a Stackelberg equilibrium with a worst-case approach. We investigate a general model in the spirit of Cvitani\'c, Possama\"i and Touzi (2018). We show that optimal contracts depend on the output and its quadratic variation, as an extension of the works of Mastrolia and Possama\"i (2016) (by dropping all the restrictive assumptions) and Sung (2015) (by considering a general class of admissible contracts). We characterize the best reaction effort of the agent through the solution to a second order BSDE and we show that the value of the problem of the Principal is the viscosity solution of an Hamilton-Jacobi-Bellman-Isaacs equation, without needing a dynamic programming principle, by using stochastic Perron's method.