Universal approximation by signatures for infinite-dimensional rough paths

TL;DR

This paper proves that functions on infinite-dimensional rough paths can be universally approximated by signature-based functions, providing a foundation for applications like stochastic PDEs and financial modeling.

Contribution

It establishes universal approximation theorems for infinite-dimensional rough paths using signature functions, extending prior finite-dimensional results to more complex settings.

Findings

Universal approximation holds in both norm and weak* topologies.

Approximation is uniform on norm-bounded sets in the weak* topology.

Lays groundwork for applications in stochastic PDEs and finance.

Abstract

We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric H\"older continuous rough paths can be approximated by functions that are linear in the signature of the path. The underlying topology determining continuity and compactness can be either the norm topology or the weak$^*$ topology. Whereas considerably more effort is required to obtain the universal approximation theorem with respect to the weak$^*$ topology, this setting ensures uniform approximation on norm-bounded sets. The motivation for establishing universal approximation theorems lies in the desire to approximate quantities derived from the solution of a stochastic partial differential equation. More specifically, our universal approximation theorems form the foundations of a novel approach to e.g. pricing of forward rates within the Heath--Jarrow--Morton--Musiela framework.