Extend rough optimal control results to general p-variation (p ≥ 3)

Extend the dynamic programming principle, the rough Hamilton–Jacobi–Bellman equation, and the verification theorem for pathwise optimal control of rough differential equations driven by geometric p-rough paths from the regime 2 ≤ p < 3 to general p-variation with 3 ≤ p < ∞. Specifically, develop a rigorous theory that covers controlled dynamics of the form dX_s = b(X_s, γ_s) ds + λ(X_s, γ_s) dζ_s with control evolution dγ_s = h(γ_s, u_s) ds, and prove that the associated value function is characterized by a well-posed rough HJB equation and admits a verification theorem in this higher p-variation setting.

Background

The paper establishes the dynamic programming principle, a rough Hamilton–Jacobi–Bellman equation, and a verification theorem for rough optimal control in the case of p-variation with 2 ≤ p < 3. This regime corresponds to rough paths where second-level iterated integrals suffice, aligning with many stochastic processes of interest such as Brownian motion.

For general p-variation with 3 ≤ p < ∞, additional algebraic and analytic complexities arise due to higher-level iterated integrals and more intricate controlled rough path structures. The authors explicitly note that extending the control-theoretic results to this broader class remains unresolved, identifying it as a central theoretical challenge.