Rough Path Approaches to Stochastic Control, Filtering, and Stopping (2509.03055v1)

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.

• The paper introduces a unified framework that replaces classical probabilistic methods with rough path techniques to handle irregular signals.
• It extends verification theorems using controlled rough paths and rough differential equations to improve optimal control under uncertainty.
• The approach employs signature methods in filtering and stopping, providing near-optimal strategies in noisy, uncertain environments.

Rough Path Approaches to Stochastic Control, Filtering, and Stopping (2509.03055)

Introduction

This paper introduces a unified framework for applying rough path theory to three classical problems in stochastic analysis: optimal control, robust filtering, and optimal stopping. The traditional probabilistic methods employed in these fields are often inadequate for handling signals with irregular paths, such as those driven by Brownian motion or environments with market microstructure noise. Instead, Lyons' theory of rough paths and Gubinelli's controlled rough paths provide a pathwise framework that generalizes classical stochastic calculus to accommodate low-regularity signals.

Rough Path Preliminaries

Rough path theory extends integration in a way that accommodates non-semimartingale inputs, where stochastic calculus based on Itô's framework fails. The theory captures the algebraic structure of iterated integrals, enabling the redefinition of integration and differential equations for irregular processes. A rough path is characterized by a path's \textit{lift}, consisting of coordinate paths and higher-order iterated integrals, which adhere to Chen's relations.

Stochastic Control in Rough Paths

General Integration Theory for Rough Paths: The rough integral's construction is similar to classical integration but employs controlled path derivatives to manage irregular processes.

Optimal Control Framework: Control problems involving irregular signals demand novel strategies. Systems leveraging rough differential equations (RDEs), driven by rough paths, have replaced traditional SDE-based models, providing robustness in evolving scenarios.

Verification Theorem: A pivotal result in control theory, this paper extends the verification theorem to rough path settings with $p$-variation, solidifying the connection between potential solutions and the Hamilton–Jacobi–BeLLMan (HJB) equation.

Robust Filtering via Rough Paths

Filtering classical signal models, such as the Kalman-Bucy framework, face limitations under parameter uncertainty. Leveraging pathwise control, rough paths reformulate the filtering process to maximize robust estimation, employing convex expectations that penalize model misfit.

Robust filters optimize expectations over model likelihoods, navigating parameter uncertainty and refining conventional estimations.

Optimal Stopping with Signature Methods

Stopped Rough Paths and Randomized Times: The paper transitions classical stopping times to measurable functions of augmented rough paths, linking them to signature elements for better inferential structures.

Linear Signature Policies: Using path signatures to capture dynamic processes' essence allows efficient computation and assessment of linear stopping policies, offering near-optimal solutions within this deterministic setting.

Conclusion

By addressing core problems like optimal control, filtering, and stopping, the unified approach presented in this paper adapts to stochastic systems' inherent irregularities. Future directions might include broadening $p$-variation settings and refining numerical methods, empowering further robust applications across stochastic modeling domains.