Universal approximation with signatures of non-geometric rough paths

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.

💡 Research Summary

This paper establishes a universal approximation theorem for signatures of rough paths that are not required to be weakly geometric. The authors achieve this by augmenting the original path with two additional components: the deterministic time variable and the rough‑path bracket (essentially the pathwise quadratic variation). The resulting extended path, denoted (t, X, ⟦X⟧), can be lifted to a genuine rough path via Lyons’ extension theorem. Although the original rough path may be non‑geometric, the extended signature satisfies a “quasi‑shuffle” property, which is sufficient to guarantee that the linear span of its components separates points. Consequently, linear functionals of the signature form a dense sub‑algebra in the space of continuous functionals on compact subsets of the rough‑path space, yielding the desired universal approximation result (Theorem 2.8).

Beyond the deterministic setting, the authors introduce the notion of a γ‑signature. They work with continuous paths of finite p‑variation (p∈(2,3)) that satisfy a regularity condition called γ‑RIE, recently studied in the literature. Using a generalized Föllmer‑type pathwise stochastic integral, they define pathwise Itô, Stratonovich, and backward‑Itô integrals in a unified way. The γ‑signature is then defined as Lyons’ extension of the canonical rough path obtained from these integrals. Corollary 3.10 shows that linear functionals of the γ‑signature (again with time and bracket augmentation) enjoy the same universal approximation property. This framework unifies deterministic and stochastic integration, allowing one to treat Itô‑type signatures on the same footing as classical geometric signatures.

In the probabilistic realm, the paper translates the deterministic results to continuous semimartingales. By constructing the Itô signature—i.e., the collection of iterated Itô integrals—of a semimartingale X and augmenting it with time and the quadratic variation ⟨X⟩, the authors prove a universal approximation theorem for linear functionals of this extended Itô signature (Corollary 4.5). This removes the need for the weak‑geometric assumption that is typically required when using Stratonovich integrals, thereby providing a rigorous foundation for directly employing Itô signatures in financial mathematics, where the martingale property is essential.

The practical relevance of the theory is demonstrated through several numerical experiments in Section 5. The authors calibrate models to real financial time‑series using features derived from time‑augmented and quadratic‑variation‑augmented signatures. They then apply these features to payoff approximation and option pricing problems, including both European and American options whose payoffs naturally depend on the quadratic variation (e.g., variance swaps). The experiments show that incorporating the quadratic‑variation component improves predictive accuracy and yields more stable parameter estimates compared with signatures that only include time. Moreover, the Itô‑based signatures capture volatility‑smile effects more faithfully than their Stratonovich counterparts.

The paper also provides extensive technical appendices: a direct proof of the universal approximation theorem for weakly geometric signatures (Appendix A), rigorous justification of Lyons’ extension for γ‑signatures (Appendix C), and auxiliary results from rough‑path theory needed throughout the work.

In summary, the contributions are fourfold: (1) a novel universal approximation theorem for signatures of non‑geometric rough paths via time‑ and bracket‑augmentation; (2) the introduction of γ‑signatures based on a unified pathwise stochastic integral framework; (3) a probabilistic version that validates the use of Itô signatures for continuous semimartingales; and (4) concrete financial applications demonstrating the practical advantage of the extended signatures. This work substantially broadens the applicability of signature methods beyond the geometric setting and opens new avenues for their use in quantitative finance and machine learning on irregular time‑series data.