Controlled rough SDEs, pathwise stochastic control and dynamic programming principles

We study stochastic optimal control of rough stochastic differential equations (RSDEs). This is in the spirit of the pathwise control problem (Lions–Souganidis 1998, Buckdahn–Ma 2007; also Davis–Burstein 1992), with renewed interest and recent works drawing motivation from filtering, SPDEs, and reinforcement learning. Results include regularity of rough value functions, validity of a rough dynamic programming principle and new rough stability results for HJB equations, removing excessive regularity demands previously imposed by flow transformation methods. Measurable selection is used to relate RSDEs to “doubly stochastic” SDEs under conditioning. In contrast to previous works, Brownian statistics for the to-be-conditioned-on noise are not required, aligned with the “pathwise” intuition that these should not matter upon conditioning. Depending on the chosen class of admissible controls, the involved processes may also be anticipating. The resulting stochastic value functions coincide in great generality for different classes of controls. RSDE theory offers a powerful and unified perspective on this problem class.