Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universal approximation with signatures of non-geometric rough paths

Mihriban Ceylan, Anna P. Kwossek, David J. Prömel

TL;DR

This work establishes universal approximation properties for signatures of rough paths beyond the weakly geometric setting by extending paths with time and their rough-path bracket terms, enabling linear functionals of signatures to approximate continuous functionals uniformly on compacts. It then introduces γ-signatures via general pathwise stochastic integration, unifying Itô, Stratonovich, and backward Itô integrals, and proves corresponding universal approximation results in both deterministic and probabilistic frameworks. The theory is translated to continuous semimartingales, linking Itô signatures to quadratic variation, and is complemented by numerical demonstrations in model calibration and option pricing, showing practical advantages of Itô signatures when quadratic variation carries information. Overall, the paper provides a rigorous foundation for using Itô-like signatures in data-driven and financial applications, expanding the toolbox of rough-path–based feature maps with robust universality properties.

Abstract

We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of the resulting rough paths approximate continuous functionals on rough path spaces uniformly on compact sets. Moreover, we construct the signature of a path extended by its pathwise quadratic variation terms based on general pathwise stochastic integration à la Föllmer, in particular, allowing for pathwise Itô, Stratonovich, and backward Itô integration. In a probabilistic setting, we obtain a universal approximation result for linear functionals of the signature of continuous semimartingales extended by the quadratic variation terms, defined via stochastic Itô integration. Numerical examples illustrate the use of signatures when the path is extended by time and quadratic variation in the context of model calibration and option pricing in mathematical finance.

Universal approximation with signatures of non-geometric rough paths

TL;DR

This work establishes universal approximation properties for signatures of rough paths beyond the weakly geometric setting by extending paths with time and their rough-path bracket terms, enabling linear functionals of signatures to approximate continuous functionals uniformly on compacts. It then introduces γ-signatures via general pathwise stochastic integration, unifying Itô, Stratonovich, and backward Itô integrals, and proves corresponding universal approximation results in both deterministic and probabilistic frameworks. The theory is translated to continuous semimartingales, linking Itô signatures to quadratic variation, and is complemented by numerical demonstrations in model calibration and option pricing, showing practical advantages of Itô signatures when quadratic variation carries information. Overall, the paper provides a rigorous foundation for using Itô-like signatures in data-driven and financial applications, expanding the toolbox of rough-path–based feature maps with robust universality properties.

Abstract

We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of the resulting rough paths approximate continuous functionals on rough path spaces uniformly on compact sets. Moreover, we construct the signature of a path extended by its pathwise quadratic variation terms based on general pathwise stochastic integration à la Föllmer, in particular, allowing for pathwise Itô, Stratonovich, and backward Itô integration. In a probabilistic setting, we obtain a universal approximation result for linear functionals of the signature of continuous semimartingales extended by the quadratic variation terms, defined via stochastic Itô integration. Numerical examples illustrate the use of signatures when the path is extended by time and quadratic variation in the context of model calibration and option pricing in mathematical finance.
Paper Structure (19 sections, 22 theorems, 184 equations, 4 figures)

This paper contains 19 sections, 22 theorems, 184 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The signature of rough paths
  3. Algebraic setting for signatures
  4. Essentials on rough path theory
  5. Definition and properties of signatures
  6. Universal approximation with signatures of general rough paths
  7. Discussion on approximation with weakly geometric rough paths
  8. The signature using general pathwise stochastic integration
  9. On Property γ-(RIE)
  10. The γ-signature
  11. A universal approximation theorem with γ-signatures
  12. Application to continuous semimartingales
  13. Numerical examples
  14. Calibration of signature models
  15. Payoff approximation and pricing
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $p \in (2,3)$. Let $(X,\mathbb{X}^{(2)}) \in \mathcal{C}^p([0,T];\mathbb{R}^d)$ be a continuous rough path such that $\mathbb{S}(\mathbb{X}^{(2)}_{0,t}) = \frac{1}{2} X_{0,t} \otimes X_{0,t}$, $t \in [0,T]$, where we consider the decomposition into the symmetric and the antisymmetric part given where $(\cdot)^\top$ denotes matrix transposition. Then $\mathbb{X}^{2}$ is a weakly geometric roug

Figures (4)

  • Figure 1: Regression on the price of the Heston model as defined above.
  • Figure 2: Regression on the price of a singular time-changed SDE as defined above.
  • Figure 3: Signature-based payoff regression and pricing under the two-asset Heston model as defined above.
  • Figure 4: Signature-based payoff regression and pricing under the two-asset singular-time changed SDE described above.

Theorems & Definitions (59)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6: Quasi-shuffle property
  • Remark 2.7
  • proof
  • ...and 49 more