Paper: arXiv 2509.03055

Authors: Jonathan A. Mavroforas, Anthony H. Dooley

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly sophisticated theoretical framework unifying optimal control, filtering, and stopping via rough path theory, featuring advanced mathematical proofs such as the central verification theorem. However, it is entirely pure/theoretical math finance with no empirical data, backtesting, or computational implementations.

Research Flowchart

  flowchart TD
    A[Goal: Unified Pathwise Framework for Control, Filtering & Stopping] --> B[Inputs: Lyons' Rough Paths & Gubinelli's Controlled Paths]
    B --> C[Methodology: Recast Classical Stochastic Problems Deterministically]
    C --> D[Computational Core: Formulate Rough HJB & Pathwise Equations]
    D --> E[Key Outcome: Rigorous Proof of Central Verification Theorem]
    E --> F[Future Work: Extend Framework to General p-Variation]

Cited References (18)

  1. Pathwise Stochastic Control with Applications to Robust Filtering (2019) — link
  2. A course on rough paths (2020)
  3. Controlling Rough Paths (2004) — link
  4. Stochastic control with rough paths (2017)
  5. Advanced Calculus (2014)
  6. Real and Complex Analysis (1987)
  7. Stochastic controls: Hamiltonian systems and HJB equations (1999)
  8. On the Bellman equation for some unbounded control problems (1997)
and 10 more
  1. Fundamentals of Stochastic Filtering (2009)
  2. Stochastic Differential Equations: An Introduction with Applications (2013)
  3. Parameter uncertainty in the Kalman–Bucy filter (2019)
  4. Differential Equations Driven by Rough Paths (2007)
  5. The structure of measurable mappings on metric spaces (1994)
  6. Optimal execution with rough path signatures (2020)
  7. Optimal stopping with signatures (2023)
  8. Algorithmic and High-Frequency Trading (2015)
  9. Numerical Methods for Approximating Solutions to Rough Differential Equations (2008) — link
  10. Runge-Kutta methods for rough differential equations (2020) — link

0 of 18 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.