# Global universal approximation with Brownian signatures

arXiv: https://arxiv.org/abs/2512.16396 · HTML: https://mlsanity.com/paper/2512.16396/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 35 references that are in the map.
- First posted on arXiv 18 Dec 2025.

## Builds on (most cited first, 10 of 10)

- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [Signature Methods in Machine Learning](https://mlsanity.com/paper/2206.14674/) (2022)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Sparse arrays of signatures for online character recognition](https://mlsanity.com/paper/1308.0371/) (2013)
- [Persistence paths and signature features in topological data analysis](https://mlsanity.com/paper/1806.00381/) (2018)
- [Signature Methods in Stochastic Portfolio Theory](https://mlsanity.com/paper/2310.02322/) (2023)
- [Deep Signature Algorithm for Multi-dimensional Path-Dependent Options](https://mlsanity.com/paper/2211.11691/) (2022)
- [Stochastic Control with Signatures](https://mlsanity.com/paper/2406.01585/) (2024)

## Built on by (most cited first, 2 of 2)

- [Weighted universal approximation of differentiable maps on infinite-dimensional manifolds](https://mlsanity.com/paper/2606.09820/) (2026)
- [Global universality via discrete-time signatures](https://mlsanity.com/paper/2603.09773/) (2026)

## Abstract (arXiv)

We establish Lp-universal approximation theorems for general path-dependent 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 the Lp-distance. To that end, we derive global universal approximation theorems for weighted rough path spaces. We demonstrate that these Lp-universal approximation theorems apply to Gaussian processes, in particular, to fractional 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.