Pontryagin Maximum Principle for rough stochastic systems and pathwise stochastic control

We analyze a novel class of rough stochastic control problems that allows for a convenient approach to solving pathwise stochastic control problems with both non-anticipative and anticipative controls. We first establish the well-posedness of a class of controlled rough SDEs with affine rough driver and establish the continuity of the solution w.r.t.~the driving rough path. This allows us to define pathwise stochastic control problems with anticipative controls. Subsequently, we apply a flow transformation argument to establish a necessary and sufficient maximum principle to identify and characterize optimal strategies for rough and hence pathwise stochastic control problems. We show that the rough and the corresponding pathwise stochastic control problems share the same value function. For the benchmark case of linear-quadratic problems with bounded controls a similar result is shown for optimal controls.