Global universal approximation with Brownian signatures

2025-12-18

AI Summary

This paper addresses the approximation of non-anticipative stochastic functionals under Brownian filtration. Methodologically, it introduces a linear functional construction based on time-extended Brownian path signatures. Theoretically, it establishes, for the first time, a global $L^p$-type universal approximation theorem within the weighted rough path space, rigorously proving that such linear functionals form a dense subset in the $L^p$-norm for all $p$-integrable adapted processes—including solutions to stochastic differential equations. By integrating rough path theory, path signature analysis, and time-extension techniques, the approach overcomes the classical anticipativity assumption, thereby extending universal approximation to the full class of non-anticipative stochastic functionals. This result provides the first rigorous, non-asymptotic functional approximation framework for data-driven modeling and deep learning of stochastic processes.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Establishes L^p universal approximation for rough path functionals
• Shows linear functionals on Brownian signatures approximate stochastic processes
• Applies to solutions of stochastic differential equations via signatures

Innovation

• Linear functionals approximate using Brownian motion signatures
• Global universal theorems for weighted rough path spaces
• Approximation of stochastic processes via time-extended signatures