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Universal approximation by signatures for infinite-dimensional rough paths

Sonja Cox, Asma Khedher, Thijs Maessen

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 Hölder 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.

Universal approximation by signatures for infinite-dimensional rough paths

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 Hölder 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.
Paper Structure (29 sections, 40 theorems, 228 equations)

This paper contains 29 sections, 40 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Outlook
  3. Structure of the article
  4. Notation
  5. A brief introduction to signatures and rough paths in infinite dimensions
  6. The tensor algebra and multiplicative functionals
  7. Hölder norms and the Hölder-metric
  8. The Lyons lift and the signature
  9. Weakly geometric rough paths
  10. Time-extended rough paths
  11. Main results: universal approximation theorems for Banach space valued geometric rough paths
  12. An abstract UAT for geometric rough paths based on Stone--Weierstrass
  13. A UAT for norm-compact sets of geometric rough paths
  14. A UAT for norm-bounded geometric rough paths
  15. The Banach space of Hölder continuous functions and its predual
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $E$ be a real Banach space, let $\tau$ be a Hausdorff topology on $\mathcal{C}^{\alpha}([0,T];E)$, let $K$ be a $\tau$-compact set, and let $D$ be a subspace of $\bigoplus_{n=0}^\infty (E^*)^{\otimes_a n}$ such that Then for any $\tau$-continuous function $f\colon K \to {\mathbb{R}}$ and for all $\epsilon>0$, there exists $l \in D$ such that

Theorems & Definitions (128)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: Tensor algebra, see e.g. Definition 2.4 in lyons_differential_2007
  • Definition 2.4: Truncated tensor algebra, see e.g. Definition 2.5 in lyons_differential_2007
  • Remark 2.5
  • Definition 2.6: Multiplicative functionals, see e.g. Definition 3.1 in lyons_differential_2007
  • Definition 2.7
  • Proposition 2.8: From Section 2.2.1 of lyons_differential_2007
  • proof
  • ...and 118 more