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 Holmström and Milgrom by combining a Stackelberg equilibrium with a worst-case approach. We investigate a general model in the spirit of [J. Cvitanić, D. Possamaï, and N. Touzi, Management Sci., 63 (2016), pp. 3328--3346], [J. Cvitanić, D. Possamaï, and N. Touzi, Finance Stoch., 22 (2018), pp. 1--37]. We show that optimal contracts depend on the output and its quadratic variation by extending [T. Mastrolia and D. Possamaï, J. Optim. Theory Appl., 179 (2018), pp. 452--500], [J. Sung, SSRN Electronic Journal, 1 (2015), 2601174]. We compute the optimal 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 a Hamilton--Jacobi--Bellman--Isaacs equation, by using the stochastic Perron's method.