Paper: arXiv 2603.09773

Authors: Mihriban Ceylan, David J. Prömel

Abstract

We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms, under an integrability condition on the underlying weight function. As an application, we show that piecewise linear interpolations of Brownian motion satisfies this integrability condition. Consequently, we obtain $L^p$-approximation results for path-dependent functionals, random ordinary differential equations, and stochastic differential equations driven by Brownian motion.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework establishing global universal approximation theorems for discrete-time path signatures using weighted function spaces and rough path theory. However, it is entirely theoretical with no empirical backtests, empirical datasets, or practical quantitative implementations included.

Research Flowchart

  flowchart TD
    A["Research Goal: Global Universal Approximation<br>for Path-Dependent Functionals"] --> B["Input: Piecewise Linear Paths &<br>Signature Features"]
    B --> C["Establish Weight Function Integrability<br>Condition on Path Spaces"]
    C --> D["Apply Framework to PL Interpolations<br>of Brownian Motion"]
    D --> E["Prove Density of Linear Signature Functionals<br>under L^p & Weighted Norms"]
    E --> F["Outcomes: L^p Approximations for Path Functionals,<br>Random ODEs & SDEs"]