Global universal approximation with Brownian signatures

Paper: arXiv 2512.16396

Authors: Mihriban Ceylan, David J. Prömel

Abstract

We establish $L^p$-type universal approximation theorems for general and non-anticipative functionals on suitable rough path spaces, showing that linear functionals acting on signatures of time-extended rough paths are dense with respect to an $L^p$-distance. To that end, we derive global universal approximation theorems for weighted rough path spaces. We demonstrate that these $L^p$-type universal approximation theorems apply in particular to Brownian motion. As a consequence, linear functionals on the signature of the time-extended Brownian motion can approximate any $p$-integrable stochastic process adapted to the Brownian filtration, including solutions to stochastic differential equations.

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 is a purely theoretical paper in mathematical finance and stochastic analysis establishing L^p-type universal approximation theorems for Brownian signatures. It features advanced, dense measure-theoretic and rough path mathematics with full proofs, but completely lacks empirical data, backtests, or code implementation.

Research Flowchart

  flowchart TD
    A[Research Goal: Global L^p Universal Approximation for Path Functionals] --> B[Inputs: Time-Extended Rough Paths & Brownian Motion]
    B --> C[Derive Global Universal Approximation on Weighted Rough Path Spaces]
    C --> D[Extend Analysis to Non-Anticipative Functionals & L^p Distances]
    D --> E[Apply Framework to Brownian Filtration]
    E --> F[Key Outcome: Linear Signatures Density for Stochastic Processes & SDE Solutions]