Global universal approximation with Brownian signatures

TL;DR

This paper proves that linear functionals on signatures of Brownian motion can universally approximate any $p$-integrable adapted process, extending the universal approximation concept to rough path spaces and stochastic processes.

Contribution

It establishes $L^p$-type universal approximation theorems for non-anticipative functionals on rough path spaces, including Brownian motion, using signatures and weighted rough path spaces.

Findings

Linear functionals on signatures are dense in $L^p$-distance.

Applicable to approximation of solutions to stochastic differential equations.

Universal approximation holds for Brownian motion in rough path framework.

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.