Pathwise Optimal Control and Rough Fractional Hamilton-Jacobi-Bellman Equations for Rough-Fractional Dynamics
Andrea Iannucci, Dan Crisan, Thomas Cass
TL;DR
This work develops a pathwise optimal control framework for systems driven by rough paths augmented with fractional dynamics. By introducing a fractional pseudo-control via $D^{\alpha}_{0^+}(\gamma - a)=u$ and restricting controls to AC^{\alpha} (with $\alpha> p/\lfloor p\rfloor$), the authors address degeneracy and define a well-posed rough fractional control problem. They establish a Dynamic Programming Principle, provide stability and Lipschitz bounds for the controlled rough differential equations, and prove well-posedness of a rough fractional HJB equation by constructing a ci-differentiable viscosity framework and analyzing a bounded-variation approximated problem. The framework is illustrated with a toy insider-trading model and connects to Gomoyunov’s fractional viscosity theory, extending prior pathwise control results to a broader class of controls and noise terms with fractional dynamics.
Abstract
We use a rough path-based approach to investigate the degeneracy problem in the context of pathwise control. We extend the framework developed in arXiv:1902.05434 to treat admissible controls from a suitable class of Hölder continuous paths and simultaneously to handle a broader class of noise terms. Our approach uses fractional calculus to augment the original control equation, resulting in a system with added fractional dynamics. We adapt the existing analysis of fractional systems from the work of Gomoyunov arXiv:1908.01747, arXiv:2111.14400v1 , arXiv:2109.02451 to this new setting, providing a notion of a rough fractional viscosity solution for fractional systems that involve a noise term of arbitrarily low regularity. In this framework, following the method outlined in arXiv:1902.05434, we derive sufficient conditions to ensure that the control problem remains non-degenerate.