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Global universality via discrete-time signatures

Mihriban Ceylan, David J. Prömel

Abstract

We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms, under an integrability condition on the underlying weight function. As an application, we show that piecewise linear interpolations of Brownian motion satisfies this integrability condition. Consequently, we obtain $L^p$-approximation results for path-dependent functionals, random ordinary differential equations, and stochastic differential equations driven by Brownian motion.

Global universality via discrete-time signatures

Abstract

We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to - and weighted norms, under an integrability condition on the underlying weight function. As an application, we show that piecewise linear interpolations of Brownian motion satisfies this integrability condition. Consequently, we obtain -approximation results for path-dependent functionals, random ordinary differential equations, and stochastic differential equations driven by Brownian motion.
Paper Structure (12 sections, 9 theorems, 120 equations)

This paper contains 12 sections, 9 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic setting for signatures
  4. Essentials on rough path theory
  5. Global universal approximation
  6. Universal approximation on weighted spaces
  7. Approximation of --functionals via signatures
  8. Universal approximation via discrete-time signatures of Brownian motion
  9. Approximation of path functionals
  10. Approximation of random ordinary differential equations and stochastic differential equations
  11. Approximation of random ODEs
  12. Approximation of SDEs

Key Result

Proposition 3.1

Let $\psi$ be defined as in eq: weight function. Then, the linear span of the set is dense in $\mathcal{B}_\psi(\widehat{\mathcal{C}}^\pi)$, i.e., for every map $f\in \mathcal{B}_\psi(\widehat{\mathcal{C}}^\pi)$ and every $\varepsilon>0$ there exists a linear function $\boldsymbol\ell\colon T((\mathbb{R}^{d+1}))\to\mathbb{R}$ of the form $\widehat{\mathbb{X}}^\pi_T\mapsto \boldsy

Theorems & Definitions (22)

  • Remark 2.1
  • Proposition 3.1: Universal approximation theorem on $\mathcal{B}_\psi(\widehat{\mathcal{C}}^\pi)$
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4: Universal approximation theorem on $\mathcal{B}_\psi(\Lambda_T^\pi$)
  • proof
  • Theorem 3.5: $L^p$-convergence
  • proof
  • ...and 12 more