Global universal approximation with Brownian signatures

TL;DR

The paper tackles the challenge of globally approximating path-dependent functionals on rough path spaces by leveraging signatures of time-extended rough paths. It develops $L^p$-type universal approximation theorems for both general and non-anticipative functionals using a weighted Stone–Weierstrass framework and stopped rough-path constructions. The Brownian case is treated in detail, showing that exponential moment conditions hold under Wiener measure, which yields universality of time-extended Brownian signatures to approximate any $p$-integrable Brownian-adapted process, including SDE solutions. As a result, linear functionals on Brownian signatures can approximate a broad class of stochastic dynamics, offering a rigorous foundation for signature-based models in stochastic analysis and mathematical finance.

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.

Theorems & Definitions (28)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Proposition 3.3: Universal approximation theorem on $\mathcal{B}_\psi(\widehat{C}_{d,T}^\alpha)$
• proof
• Theorem 3.4: $L^p$-universal approximation theorem on $\widehat{C}_{d,T}^\alpha$
• proof
• Remark 3.5
• Definition 3.6
• Definition 3.7