Paper: arXiv 2509.14529

Authors: Tomoyuki Ichiba, Qijin Shi

Abstract

Built to generalise classical stochastic calculus, rough path theory provides a natural and pathwise framework to model continuous non-semimartingale assets. This paper investigates the ultimate capacity of this framework to support frictionless continuous No-Free-Lunch markets à la Kreps-Yan. We establish a ``Rough Kreps-Yan" theorem, which links our No Controlled Free Lunch (NCFL) condition to the unbiasedness of the driver of the price process as a rough integrator. The central work of this paper is a complete classification of these unbiased rough integrators with respect to different classes of controlled paths as integrands. As the set of admissible trading strategies is enlarged to include Markovian-type and signature-type portfolios, the only admissible random rough paths must be infinitesimally close to the Itô rough path lift of a standard Brownian motion, up to a time change. In particular, Gaussianity is no longer a model assumption, but rather a no-arbitrage market consequence. Notably, simple strategies do not appear in the theory, and if they are then reintroduced, the rough noise is further enforced to be the Itô rough path of Brownian motion itself. Ultimately, this implies that continuous frictionless markets based on rough path theory are inevitably constrained to the semimartingale paradigm, providing a definitive answer on the limits of this approach. Our framework covers $α-$Hölder continuous rough paths for $α>0$ arbitrarily small.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a groundbreaking theoretical result establishing the structural limits of rough path theory in frictionless market models by proving a ‘Rough Kreps-Yan’ theorem. The mathematical machinery (rough differential equations, controlled paths, Hermite polynomials) is exceptionally advanced, though it is entirely theoretical with no empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Assess No-Arbitrage Limits<br>in Rough Path Market Models] --> B[Inputs: Continuous Non-Semimartingale Assets<br>& α-Hölder Rough Paths]
    B --> C[Establish 'Rough Kreps-Yan' Theorem:<br>Link No-Arbitrage NCFL to Unbiased Integrators]
    C --> D[Classify Unbiased Rough Integrators Across<br>Strategy Sets: Markovian & Signature Portfolios]
    D --> E[Key Outcome: Noise Must Formally Approach<br>Itô Lift of Brownian Motion]
    E --> F[Conclusion: Frictionless Rough Markets<br>Are Constrained to Semimartingales]

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5 of 71 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.