Paper: arXiv 2602.05898

Authors: Mihriban Ceylan, Anna P. Kwossek, David J. Prömel

Abstract

We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of the resulting rough paths approximate continuous functionals on rough path spaces uniformly on compact sets. Moreover, we construct the signature of a path extended by its pathwise quadratic variation terms based on general pathwise stochastic integration à la Föllmer, in particular, allowing for pathwise Itô, Stratonovich, and backward Itô integration. In a probabilistic setting, we obtain a universal approximation result for linear functionals of the signature of continuous semimartingales extended by the quadratic variation terms, defined via stochastic Itô integration. Numerical examples illustrate the use of signatures when the path is extended by time and quadratic variation in the context of model calibration and option pricing in mathematical finance.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework extending universal approximation theorems for signatures of non-weakly geometric rough paths (Itô signatures). While highly strong on theoretical advanced probability and rough path math, the empirical section relies primarily on numerical option pricing and calibration illustrations rather than extensive backtesting on real historical datasets.

Research Flowchart

  flowchart TD
    A[Goal: Universal Approximation for Signatures of Non-Geometric Rough Paths] --> B[Input: Paths & Integration Framework]
    B --> C[Extend Paths with Time & Quadratic Variation Terms]
    C --> D[Construct Extended Signatures via Föllmer & Itô Integration]
    D --> E[Prove Linear Functionals Uniformly Approximate Continuous Functionals]
    E --> F[Numerical Application: Model Calibration & Option Pricing in Finance]
    F --> G[Outcome: Extended Signatures Achieve Universal Approximation]