The Pontryagin maximum principle and Q-functions in rough environments

We derive the Pontryagin maximum principle and Q-functions for the relaxed control of noisy rough differential equations. Our main tool is the development of a novel differentiation procedure along `spike variation' perturbations of the optimal state-control pair. We then exploit our development of the infinitesimal Q-function (also known as the q-function) to derive a policy improvement algorithm for settings with entropic cost constraints.