Pontryagin Maximum Principle for rough stochastic systems and pathwise stochastic control
Ulrich Horst, Huilin Zhang
TL;DR
The paper develops a rigorous framework for rough stochastic control with affine rough drivers and its pathwise, anticipative counterpart. It proves well-posedness and Itô–Lyons continuity for the controlled rough SDEs, and uses a rough Doss–Sussmann transformation to derive a Pontryagin-type maximum principle, establishing equivalence between rough and pathwise control problems and their value functions. The analysis encompasses both non-anticipative and anticipative controls, and extends to generalized pathwise problems with a focus on measurability and LQ examples. The results broaden the scope of stochastic control by enabling robust pathwise formulations and providing tools for mean-field and robust control applications where exogenous noise can be observed or structured as a rough path.
Abstract
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.