Rough Path Approaches to Stochastic Control, Filtering, and Stopping
Abstract
This paper presents a unified exposition of rough path methods applied to optimal control, robust filtering, and optimal stopping, addressing a notable gap in the existing literature where no single treatment covers all three areas. By bringing together key elements from Lyons' theory of rough paths, Gubinelli's controlled rough paths, and related developments, we recast these classical problems within a deterministic, pathwise framework. Particular emphasis is placed on providing detailed proofs and explanations where these have been absent or incomplete, culminating in a proof of the central verification theorem, which is another key contribution of this paper. This result establishes the rigorous connection between candidate solutions to optimal control problems and the Hamilton-Jacobi-Bellman equation in the rough path setting. Alongside these contributions, we identify several theoretical challenges -- most notably, extending the verification theorem and associated results to general p-variation with -- and outline promising directions for future research. The paper is intended as a self-contained reference for researchers seeking to apply rough path theory to decision-making problems in stochastic analysis, mathematical finance, and engineering.